'Pitt O.A.C. ELECTRIC POWER TRANSMISSION PUBLISHERS OF BOOKS F O R^. Electrical World v Engineering News-Record Power v Engineering and "Mining Journal-Press Chemical and Metallurgical Engineering Electric Railway Journal v Coal Age American "Machinist v Ingenieria Internacional Electrical Merchandising ^ BusTransportation Journal of Electricity and Western Industry Industrial Engineer ELECTRIC POWER TRANSMISSION PRINCIPLES AND CALCULATIONS INCLUDING A REVISION OF "OVERHEAD ELECTRIC POWER TRANSMISSION" BY ALFRED STILL PROFESSOR OF ELECTRICAL DESIGN, PURDUE UNIVERSITY; MEMBER OF THE INSTITUTION OF ELECTRICAL ENGINEERS; FELLOW OF THE AMERICAN INSTITUTE OF ELECTRICAL ENGINEERS; MEMBER OF THE INSTITUTION OF CIVIL ENGINEERS; AUTHOR OF " POLYPHASE CURRENTS;" " PRINCIPLES OF ELECTRICAL DESIGN;" ETC. SECOND EDITION REVISED, ENLARGED AND REWRITTEN FIFTH IMPRESSION McGRAW-HILL BOOK COMPANY, INC. NEW YORK: 370 SEVENTH AVENUE LONDON: 6 & 8 BOUVERIE ST., E. C. 4 1919 COPYRIGHT, 1913, 1919, BY THE MCGRAW-HILL BOOK COMPANY, INC. PRINTED IN THE UNITED STATES OF AMERICA TK PREFACE TO SECOND EDITION When this book was published originally under the title "Overhead Electric Power Transmission," its suitability for use as a College text had not been seriously considered. It has, however, been used by a large number of technical schools and colleges, and while the restricted scope of the book may limit its suitability as a text for college students, the changes and additions which will be found in this new edition should, in the author's opinion, enhance its value as a college text without detracting from its usefulness in the field of practical engineering. The principal addition which has necessitated an alteration in the title is an entire chapter treating of Underground Con- ductors. This has been written with the kind assistance of Mr. C. J. Beaver who has not only furnished most of the data relating to underground cables, but has also read and criticised the matter presented in Chapter VII. A portion of the material which was originally contained in the Appendix has been incorporated in the text; but much of the first edition has been entirely omitted, either because its inclusion is no longer necessary owing to the rapid strides that have been made of late years in the general knowledge of electrical power transmission, or because it has been replaced by new material believed to be of more value to the student or practical engineer. The chapter describing the Thury system of transmission by continuous currents has been retained with only slight changes and additions. It has not been deemed expedient to omit this entirely because, although few American engineers have taken the trouble to familiarize themselves with this system of trans- mission, there are conditions under which it has certain in- disputable advantages which European engineers have not been slow to recognize. The costs, both of material and labor, which are given in Chapter III, are not representative of market conditions on or about the date of publication of this book. Their principal use is to give an idea of the relative costs of different parts of a transmission system. They are based on trade conditions prevailing during the two or three years immediately preceding the war. A. STILL. PURDUE UNIVERSITY, LAFAYETTE, INDIANA, June, 1919. PREFACE TO FIRST EDITION Although this book treats mainly of the fundamental principles and scientific laws which determine the correct design of over- head electric transmission lines, it has been written primarily to satisfy the needs of the practical engineer. An attempt has been made to give the reasons of things to explain the deriva- tion of practical methods and formulas in the simplest possible terms: the use of higher mathematics has been avoided; but vector diagrams, supplemented where necessary with trigono- metrical formulas, have been freely used for the solution of alter- nating-current problems. It is therefore hoped that the book may prove useful, not only to the practical designer of trans- mission lines, but also to those engineering students who may wish to specialize in the direction of Power Generation and Transmission, for these will find herein a practical application of the main theoretical principles underlying all Electrical Engineering. The subject is treated less from the standpoint of the construc- tion engineer in charge of the erection work, as of the office engineer whose duty it is to make the necessary calculations and draw up the specifications. The considerations and practi- cal details of special interest to the engineer in charge of the work in the field have already been presented in admirable form by Mr. R. A. Lundquist in his book on Transmission Line Construction. Much of what appears in these pages is reprinted with but little alteration from articles recently contributed by the writer to technical journals; but in the selection and co-ordination of this material, the scheme and purpose of the book have steadily been kept in mind. Systems of distribution, whether in town or country, are not touched upon: the subjects dealt with cover only straight long- distance overhead transmission. It is true that, when treating of lightning protection, it is the machinery in the station buildings rather than the line itself that the various devices referred to are intended to protect; and, when considering the most economical viii PREFACE system of transmission under given circumstances, a thorough knowledge of the requirements and possibilities in the arrange- ment of generating and transforming stations is assumed; but these engineering aspects of a complete scheme of power develop- ment are not included in the scope of this book. In the Appendix will be found reprints of some articles dealing with theoretic aspects of long-distance transmission which, although believed to be of interest to anyone engaged on the design of transmission lines, are not essential to the scheme of the book. In the Appendix will also be found complete speci- fications for a wood pole and steel tower line respectively: these should be helpful, not so much as models for other specifications every engineer is at liberty to draw these up in his own way but rather as containing suggestions and reminders that may be of service when specifying and ordering materials for an actual overhead transmission. The writer desires to thank the editors of the following tech- nical journals for permission to reprint articles or portions of articles which they have published from time to time: Electrical World, New York; Electrical Times, London; Canadian Engineer, Toronto; Western Engineering, San Francisco; Journal oj Elec- tricity, Power, and Gas, San Francisco. PURDUE UNIVERSITY, LAFAYETTE, INDIANA, August, 1913. CONTENTS PAGE Preface to Second Edition v Preface to First Edition vii List of Symbols xv CHAPTER I Introductory and General 1 CHAPTER II Electrical Principles and Theory Elementary 1. Losses in Transmission 13 2. Transmission by Continuous Currents 14 3. Transmission by Single-phase Alternating Currents 14 4. Transmission by Two-phase Currents 15 6. Transmission by Three-phase Currents 17 6. Relative Cost of Conductors Required on the Various Systems. . . 19 7. Grounding the Neutral on High-tension Overhead Transmissions . . 23 8. Regulation: Effect of Line Inductance on the Transmission of Al- ternating Currents 24 9. Fundamental Vector Diagram for Line Calculations: Capacity Neglected 25 10. Effect of Capacity on Regulation and Line Losses 28 11. Use of Fundamental Diagram for Three-phase Calculations 32 CHAPTER III Economic Principles and Calculations 12. Introductory 36 13. Choice of System 37 14. Type of Transmission Line 37 15. Length of Span 38 16. Effect of Span Variations on Cost of Steel Towers 39 17. Cost of Wood Poles 41 18. Cost of Insulators 42 19. Duplicate Lines 42 20. Costs of Typical Transmission Lines 43 21. Cost of Overhead Conductors 48 22. Economic Size of Conductor. Kelvin's Law 49 ix x CONTENTS PAGE 23. Practical Method of Applying Kelvin's Law 51 24. Economic Ohmic Voltage Drop 52 25. Economic Voltage, and Calculation of Conductor Sizes 53 26. Example Illustrating Quick Method of Determining Economic Size of Conductors 55 27. Estimation of Amount and Cost of Energy Wasted in Conductors. 57 28. Estimation of Percentage to Cover Annual Interest and Depre- ciation on Conductors 58 29. Economic Voltage 59 30. Costs Other than Transmission Line, Liable to be Influenced by Voltage Variations 60 31. Annual Charges Depending on Voltage 60 32. Depreciation 61 33. Example: Method of Determining Most Economical Voltage 63 34. Closer Estimate of Economical Voltage 66 CHAPTER IV Electrical Principles and Calculations 36. Materials 69 36. Copper 70 37. Aluminum 70 38. Iron and Steel 71 39. Copper-clad Steel 72 40. Stranded Cables with Steel Wire Core 73 41. Physical Constants and Sizes of Commercial Conductors 74 42. Skin Effect 77 43. Inductance of Transmission Lines 79 44. Effect of Taking into Account the Return Conductor. 80 45. Effect of Flux Lines in the Material of the Conductor 82 46. Iron as a Material for Transmission Line Conductors 83 47. Apparent Resistance of Iron and Steel Conductors 84 48. Internal Reactance of Iron and Steel Conductors 85 49. Example of Calculations of Iron Wire Conductors 88 60. Inherent Regulation of Transmission Line. Regulation Diagrams . 89 61. Pressure Available at Intermediate Points on a Transmission Line. . 94 62. Capacity of Transmission Lines 96 53. Capacity of Three-phase Lines 97 64. Charging Current Due to Capacity of Transmission Lines 99 56. Effect of Distributed Capacity and Inductance .101 56. Electrical Calculation of Lines with Appreciable Capacity 102 67. Numerical Examples Illustrating Use of Formulas for the Cal- culation of Power Factor and Voltage Drop 104 68. Distinction Between Regulation and Line Drop 108 69. Line Losses 110 60. Control of Voltage on Transmission Lines 112 61. Effect of Boosting Voltage at Intervals along a Transmission Line. 114 CONTENTS xi PAGE 62. Control of Power Factor. 115 63. Use of Rotary Reactors to Control the Voltage 117 64. Power Factor of Load 120 65. Grounded versus Isolated Transmission Systems 120 66. Interference between Power and Telephone Lines 122 67. Insulation of Telephone Lines 122 68. Electrostatic Induction 122 69. Magnetic Induction 123 70. Fault Localizing 125 CHAPTER V Insulation of Overhead Transmission Lines 71. Insulator Materials 127 72. Design of Insulators 129 73. Pin-type Insulators 131 74. Suspension-type Insulators 134 75. Wall and Roof Outlets 143 76. Design of Insulating Bushings 146 77. Condenser Type of Bushing 150 78. Formation of Corona, and Accompanying Losses of Power 151 79. Corona Considered as "Safety Valve" for relief of High-frequency Surges or Over-voltage Due to Any Cause 156 80. Spacing of Overhead Conductors 156 81. Practical Limitations of Overhead Transmission-line Voltages 158 82. Factors of Safety: Rating and Testing of Line Insulators 159 CHAPTER VI Protection against Lightning Transient Phenomena 83. Theoretical Considerations 163 84. Frequency of Oscillations 167 85. Wave Length 168 86. Reflection of Travelling Waves 170 87. Line Disturbances Caused by Switching Operations 173 88. Lightning 174 89. Protection of Overhead Systems against Direct Lightning Strokes and Sudden Accumulations of High Potential Static Charges. 175 90. Protection of Insulators from Power Arcs 177 91. Methods of Grounding 178 92. Relieving Conductors of High Potential "Static." Water Jet Arresters 179 93. Horn Gap 180 94. Multiple-gap Low Equivalent Arrester 183 95. Spark-gap Arresters with Circuit Breakers or Re-setting Fuses. . . . 186 96. Aluminum Cell Arrester T . 187 xii CONTENTS PAGE 97. Condensers 189 98. Spacing of Lightning Arresters 192 99. Choke Coils 192 100. Arcing Ground Suppressor 194 101. General Remarks on Lightning Protection 195 CHAPTER VII Transmission of Energy by Underground Cables 102. Introductory 199 103. Submarine Power Cables 200 104. Voltage Limitations of Underground Cables 201 105. Types and Construction of Power Cables 202 106. Methods of Laying Underground Cables 206 107. Costs of Underground and Transmission Lines 208 108. Cable Terminals. Junction with Overhead Lines 211 109. Design of Cables 212 110. Economical Core Diameter of High-pressure Cables 214 111. Grading of Cables 216 112. Three-core Cables 217 113. Capacity and Charging Current of Three-core Cables 217 114. Example of Design of Single-phase Concentric E.H.T. Power Cable 221 115. Losses in Underground Cables 223 116. Temperature Rise of Insulated Cables 225 117. Reliability of Cable Systems. Joints; Electrolysis 228 CHAPTER VIII Transmission of Energy by Continuous Currents 118. General Description of the Thury System 233 119. Straight Long-distance Transmission by Continuous Currents... 236 120. Insulation of Line when Carrying Continuous Currents 237 121. Relative Cost of Conductors: Continuous Current and Three- phase Transmissions 238 122. Concluding Remarks on Direct-Current Transmission 242 CHAPTER IX Mechanical Principles and Calculations Overhead Conductors 123. Introductory 248 124. Graphical Statics Applied to Transmission-line Calculations General Problem . . 249 125. Stretched Wire. Supports on Same Level 250 126. Supports at Different Elevations , 254 CONTENTS xiii PAGE 127. Calculation of Sag with Supports on an Incline 258 128. Example Illustrating Use of Formulas 259 129. Conclusions Overhead Lines on Steep Grade 261 130. Effect of Temperature Variations on Overhead Wires 261 131. Abnormal Stresses in Wires due to Wind and Ice 264 132. Swaying of Wires in Strong Wind 273 133. Calculation of Total Stress in Overhead Wires 273 134. Effect of Temperature Variations on Sag and Stress 278 135. Calculations of Sags and Tensions under any Conditions of Load and Temperature 281 136. Tensions in Conductors when Spans are of Different Lengths 288 137. Tension in Different Sized Wires on the Same Span. 288 138. Further Example Illustrating Temperature-sag Calculations 289 139. Sag-temperature Calculations with Supports at Different Elevations 290 140. Length of Spans: Conductor Materials: Copper: Aluminum: Iron 298 141. Factors of Safety: Joints and Ties 301 CHAPTER X Transmission Line Supports 142. General Considerations: Types of Transmission Line Supports. . . 303 143. Wood Pole Lines: Kinds of Wood Available 305 144. Typical Woodpole Lines 306 145. Life of Wood Poles: Preservative Treatment 307 146. Insulating Qualities of Wood Poles 313 147. Weight of Wood Poles 313 148. Strength and Elasticity of Wood Poles 314 149. Calculation of Pole Strengths 316 150. Deflection of Wood Poles. 318 151. Calculation of Pole Deflections 318 152. Pole Foundations 320 153. Spacing of Poles at Corners: Guy Wires 321 154. Load to be Carried by Corner Poles 322 155. Props or Struts: Wood Poles in Compression 323 156. Reinforced Concrete Poles 324 157. Weight and Cost of Concrete Poles 325 158. Strength and Stiffness of Concrete Poles 327 159. Steel Poles and Towers: Introductory Remarks 329 160. Flexible Towers 331 161. Steel Poles for Small Short-distance Transmission Schemes 334 162. Loads to be Resisted by Towers 335 163. Design of Steel Towers 336 164. Stresses in Compression Members of Tower Structures 337 165. Outline of Usual Procedure for Calculating Stresses in Tower Members. . . 340 xiv CONTENTS PAGE 166. Stiffness of Steel Towers: Deflection Under Load 344 167. Tower Foundations 344 168. Concluding Remarks Regarding Steel Tower Design 347 169. Determining Position of Supports on Uneven Ground 348 170. Study of Deflections and Stresses in Flexible Tower Lines 349 171. Numerical Exam pie: Transmission Line with Flexible Supports. . 352 172. Erection of Steel Tower Transmission Lines 359 APPENDIX 1 INDUCTANCE OF TRANSMISSION LINES WITH ANY ARRANGEMENT OF PARALLEL CONDUCTORS 361 Calculation of Total Resultant Flux Surrounding one Conductor when there are Several Return Conductors Calculation of E.m.f. of Self and Mutual Induction Numerical Example: Three-phase Transmission Inductance of Electric Transmission Lines as Affected by the Subdivision of the Circuits and the Arrangement of the Conductors. APPENDIX II SPECIFICATION FOR WOOD POLE TRANSMISSION LINE 375 General Description of Transmission Line Clearing Poles Cross-arms Grading Pole Setting Grounding Spans Angles and Curves Guying Insulators Stringing Wires Locating and Numbering Poles. APPENDIX III SPECIFICATIONS FOR STEEL TOWER TRANSMISSION LINE 386 General Description of Line Duties of Engineer in Charge of Construction Clearing Towers Foundations for Towers Grounding Guying Angles Erection of Towers Insulation Conductors Joints in Conductors Spans and Wire Stringing Specification for Steel Towers Specification for Porcelain Insulators. INDEX. . . . . 397 LIST OF SYMBOLS A = area of cross section. a = temperature-elongation coefficient. a = percentage to cover annual interest and depreciation. a = n~ in capacity formulas. B = magnetic flux density (gauss). B. & S. = Brown and Sharpe wire gauge. 6 = barometric pressure: cm. or inches of mercury. C = electrostatic capacity (farad). C m = electrostatic capacity; microfarads (usually per mile of con- ductor). C e = effective equivalent capacity, core to neutral, of underground cables. D = flux density in electrostatic field (coulombs per sq. cm.). D = butt diameter of wood pole. d = distance between centers of parallel wires. d = diameter of wood pole. d g = diameter of wood pole at ground level. E = electromotive force (e.m.f.): difference of potential (volts). E = voltage between lines at receiving end. E a = voltage, line to ground, on A. C. system (in comparison with D. C.). E n voltage between wire and neutral (usually at receiving end). E = disruptive critical voltage (corona) (r.m.s. value of sine wave). E r = "economic" ohmic drop, in volts per mile of conductor. E t = visual critical voltage (corona). e = e.m.f. usually volts, e.m.f. = electromotive force. F = wind pressure: Ib. per sq. ft. / = frequency (number of periods per second). G = potential gradient = -^r (volts per centimeter). 44UU / 3000 / / ;ff / , ^; y / 2100 2200 2000 1800 1000 1200 1000 800 600 400 200 ^ . / ^ X / */ / / / > ^ y : ^ #& '> ^ ^ ^ "** These curves give the approocim cost per taile of a transmission line complete with insulators, Vot Including the cost of condu right of way, clearing -around 1 wooded country* 01 Interest on capital during con t ruction perl The labor, cost of springing the conductors iff included. ate ^ ^ bus ctors, | >a. r~ ***" 40 50 60 70 Pressure, Kilo-volts FIG. 18. Approximate cost of overhead transmission lines. and height of poles. The weight and diameter of conductors, by affecting the required strength of the supports, will be factors in determining the cost of the complete line, apart from any difference in the value of the conductors themselves. The actual cost of stringing very light or very heavy conductors will also differ from the average amount allowed for the purpose of plot- ting the curves. The number and style of lightning conductors, if any, and whether or not one or more grounded guard wires are strung above the conductors will obviously modify the average figures. Although steel poles, or steel towers, will generally be found more economical than a wood pole line for voltages above 44,000 on account of the heavier insulators, wider spacing be- tween conductors, and generally greater height of support, it 48 ELECTRIC POWER TRANSMISSION does not follow that wood poles or wood-pole structures may not prove economical, even for comparatively high voltages, in coun- tries where suitable timber is plentiful and the ready means of transportation and erection of steel towers are wanting. For instance, the cost per mile, as obtained from Fig. 18, is $2600 for a single 100,000-volt three-phase line supported on rigid square base steel towers, and it is not improbable that an entirely satis- factory wood pole line could be built under favorable conditions at a figure appreciably below $2000 per mile (not including conductors, right-of-way, or clearing wooded land). But it does not follow that the wood pole line is the most economical, because the probable cost of maintenance, repairs, replacements of de- cayed or damaged poles, and all charges to be met annually or periodically during future years must be carefully considered before a final decision can be arrived at. 21. Cost of Overhead Conductors. The capital expenditure on conductors will depend upon the material and the total weight. It is not proposed to discuss, in this place, the relative merits of copper and aluminum as conductor materials, but it may be well to point out that although the market values of these metals may be such that the use of aluminum may lead to some saving on first cost, there are many engineering points to be most care- fully considered before definitely adopting either metal. The weight of the conductors necessary to transmit a certain amount of power over a definite distance will obviously depend upon the voltage, but apart from the engineering difficulties encountered at the higher voltages, there are economic considerations which determine the maximum voltage suitable for any given conditions. Among these may be mentioned a possible increase in the cost of generating plant for the higher pressures, the greater cost of step-up and step-down transformers and of the control apparatus, together with the line insulators, entering bushings, etc. The transmission line poles or towers will also, as previously men- tioned, cost more for the higher pressures, because of the wider spacing between conductors and the greater length of insulator string. Then again, with the extra high pressures, the increased losses through leakage over insulators and possible corona losses may be quite appreciable. Given a definite amount of power to be transmitted, and a definite line pressure, the current can be calculated ; and the eco- nomic conductor cross-section and therefore the weight and ECONOMIC PRINCIPLES AND CALCULATIONS 49 cost of the conductors will be directly proportional to this current. It is only of recent years that this fact appears to have been generally recognized, and yet, so long ago as 1885, in his Cantor lectures delivered in London, Prof. George Forbes said: ''The most economical section of conductor is independent of e.m.f. and distance, and is proportional to the current." The determination of the current value to be used in the calculation of conductor sections is a real difficulty. It must not be supposed that even a knowledge of the load factor is sufficient by itself. The load factor, being the ratio of average load to maximum load, does not give the relation between the average PR loss and the I 2 R loss of maximum output. The power lost in the conductors of a constant potential supply is proportional to the square of the power transmitted. On the basis of the average hydro- electric load curve, if the load factor is 50 per cent., the load on which the average transmission line losses should be based being the square root of the mean of the square of the power transmitted- will probably be found to be more nearly 60 per cent, than 50 per cent, of the maximum load. Although the discussion which follows refers mainly to over- head transmission lines, the same general principles should govern the choice of conductor cross-section in underground cables. As an economic problem, the underground system of transmission differs from the overhead system mainly in the fact that the cost of the insulation in a cable is a function of the conductor diameter, whereas the cost of line insulators is less closely connected with the size or weight of the overhead con- ductors. The cost of insulation is relatively more important in cable systems than in overhead transmissions. Some reference to the economics of power transmission by underground cables will be made in Chapter VII. 22. Economic Size of Conductor. Kelvin's Law. Before considering to how great an extent the voltage may be raised, in order to keep down the current, without exceeding the limits determined by economic considerations, it will be well to examine, in some detail, the fundamental principle known as Kelvin's law, by which the proper size of conductor to carry a known current is determined. In this connection it is of no consequence whether the transmission is by direct or alternating currents, single phase or polyphase. If conductors have to be provided to carry a current of known amount, these may be of large cross- 50 ELECTRIC POWER TRANSMISSION section and therefore of high initial cost, but of so low a resistance that the PR losses will be small; or they may be of small cross- section and high resistance, the capital expenditure on which will be small; but in which the I*R losses will be large. The economical size of conductor for any given transmission will there- fore depend on the cost of the conductor material and the cost of the power wasted in transmission losses; and the law of maxi- mum economy may be stated as follows: The annual cost of the energy wasted per mile of the transmission line, added to the an- nual allowance (per mile) for depreciation and interest on first cost, shall be a minimum. If it is assumed that the cost of poles or towers, insulators and other materials (apart from the conductors themselves) including the labor on erection and stringing of wires, is independent of the actual size of conductor, then the only variable item in the capital expenditure is directly proportional to the cross-section (or weight) of the conductor, and since the PR losses (for a given current) are inversely proportional to the conductor cross-section, the law of maximum economy is greatly simplified, and in fact becomes Kelvin's law, which may be expressed as follows: The most economical section of a conductor is that which makes the annual cost of the I Z R losses equal to the annual interest on the capital cost of the conductor material, plus the necessary annual allowance for depreciation. The cross-section should, therefore, be determined solely by the current which the conductor has to carry, and not by the length of the line or an arbitrary limit of the percentage full-load pressure drop. If there are reasons which make a large pressure drop undesirable, then, if necessary, economy must be sacrificed, and the line calculated on the basis of regulation only. It will, however, generally be found that the economic conductor will give reasonably good regulation. The diagram, Fig. 19, shows clearly how the minimum total annual cost occurs when the cost per annum of the wasted energy is equal to the capital cost expressed as an annual charge; and if desired a graphical solution of Kelvin's law can readily be ob- tained by this means. In Fig. 19, the horizontal distances meas- ured to the right of the point O represent increasing conductor resistances; while the vertical distances represent money. The curve A shows how the annual charges depending on capital outlay decrease with increase of conductor resistance; while the straight line B indicates the growth of the cost of wasted power; ECONOMIC PRINCIPLES AND CALCULATIONS 51 this being directly proportional to the resistance. By adding the ordinates of curves A and B, the curve C is obtained, of which the lowest point indicates the resistance per mile of conductor which will be the most economical to use, whatever may be the length of the line, or the pressure required at the receiving end. .6 1.0 Resistance, Ohms per Mile, of Single Conductor FIG. 19. Graphical illustration of Kelvin's law. It will be observed that this minimum occurs where the two curves cross. 23. Practical Method of Applying Kelvin's Law. The follow- ing formulas have been evolved with a view to facilitating the calculation of conductor sizes to give the most economical 52 ELECTRIC POWER TRANSMISSION results on overhead transmissions. In every case the lesser factors which may, to a small extent, influence the results of the problem will be disregarded, but they may be taken into account when the final details of the transmission line are being con- sidered. On the other hand, it will generally be found that the application of Kelvin's law in its simplicity, without regard to such influences as the possible variations in cost of supports, in- sulators, etc., depending upon the size of the conductors, will give results sufficiently accurate for practical purposes, and this for two important reasons : 1. A small variation in the diameter of the conductor either on the large or the small side is usually of very little consequence from the economic point of view. 2. As the standard size of conductor nearest in diameter to the theoretically correct size is generally selected, refinements or increased accuracy in the calculations will rarely affect the. size of wire which is ultimately decided upon. 24. Economic Ohmic Voltage Drop. It is not generally realized that when the size of a conductor is determined by the application of Kelvin's law the ohmic drop of pressure per unit length of conductor is independent of the actual voltage or the current to be carried, and therefore bears no definite relation to the total amount of power to be transmitted. The economic data and assumptions alone determine the ohmic drop in volts per unit length of conductor, and this will be a constant quantity whatever the number of conductors or system of electric trans- mission adopted, the total amount of power to be transmitted, or the voltage ultimately decided upon. This fact very consider- ably simplifies the problem in its earlier stages. The formula for the economic voltage drop may be arrived at as follows, bearing in mind that the annual charges to be considered are (1) an annual charge for interest and depreciation on the cost of the line wire; (2) the annual cost of the energy wasted in the conductor in the form of I*R losses, 1 and that the equality of these two items of cost determines the size of the most economical conductor. Annual Charges Depending Upon Cost of Conductor. Let p be the price to be paid for 100 Ib. weight of conductor and a the 1 Other losses due to leakage over insulators and through the air should be taken into account when considering the choice of e.m.f., especially if this should exceed 60,000 volts. ECONOMIC PRINCIPLES AND CALCULATIONS 53 percentage to be taken to cover the annual interest and deprecia- tion, then, if R be the resistance in ohms per mile of the conductor, Annual charge = ~ X pX^XK (16) where K is a constant depending upon the material of the con- ductor and the temperature. Annual Cost of Energy Lost Let p\ be the cost per kw.-year of the wasted energy; then, annual cost per mile of conductor = p\ Xkw. lost per mile. where E r stands for ohmic drop in volts per mile of conductor. In order to satisfy the condition of equality between the values (16) and (17) we must write a X p X K = Pl X E r z 100 X R 1000 X R whence E r *=WK^^ (18) If the temperature is about 20 degrees Centigrade, and the material of the line is copper, the constant K may be taken as 8.76, while for aluminum it works out at 4.32. Inserting these values in the last formula, the economic ohmic voltage drop per mile of copper conductor becomes: E r = 9.35 ^ (19) and for aluminum: (20) If preferred, these formulas may be put in the form: Circular mils per ampere (copper) = 5800 \l-~- (21) \ a /\ p and, Circular mils per ampere (aluminum) = 13,400 A/ - 1 (22) 25. Economic Voltage, and Calculation of Conductor Sizes. Having ascertained what will be the most economical ohmic 54 ELECTRIC POWER TRANSMISSION drop of pressure per mile of conductor without reference to the total amount of power to be transmitted, the size of the conductor cannot be determined unless the value of the current is known, and this will depend upon the pressure at which the energy will be transmitted. If the cost of the conductors forming the transmission line, and the PR losses therein, were the only considerations, a high voltage would in all cases be desirable on account of the corre- sponding reduction of current for a given amount of energy to be transmitted. But, apart from the extra cost of the line due to the better insulation and wider spacing of wires required by the higher pressures, the cost of generation and transformation of high-pressure energy must be taken into account, and as the extra cost per kilowatt of equipment for generating at high pres- sures will depend largely upon the total output required, it follows that the most economical pressure will bear some relation to the total power to be transmitted. This is apart from the distance of transmission, which is the most important factor governing the choice of voltage. If the distance is great it is obvious that the reduction of material cost and power losses in the line due to the employment of higher pressures will be rela- tively of far greater importance than the increased cost of plant in generating and transforming stations. On the other hand, the employment of very high pressures even on a comparatively long line might not be justified if the total amount of power to be transmitted were very small. As a first approximation, the writer has found the following formula useful in getting out preliminary estimates; the line voltage given by the formula agrees generally with modern practice. I k.w. Line pressure (kilovolts) = 5.5 -\/L + -^~ (23) This empirical formula may be used for estimating the probable economical transmission voltage on lines over 20 miles in length. The symbol L stands for the distance of transmission in miles, while k.w. stands for the estimated maximum number of kilo- watts that will have to be transmitted over one pole- or tower- line. Given the amount of power to be transmitted and the length of line, one can with the aid of formula (23) decide upon a stand- ECONOMIC PRINCIPLES AND CALCULATIONS 55 ard voltage and proceed with the calculations for current and size of conductor; but it is necessary always to bear in mind that a transmission line cannot be considered by itself; it must be treated as part of a complete scheme of transmission and dis- tribution, and the best voltage to use on any given system can generally be arrived at only by a method of trial and error, taking into account the cost of the various parts of the complete system as influenced by alterations in the transmission voltage. No accurate formula can be evolved which would be applicable to all the varied conditions encountered in actual work; but a practical method of attaining the required end will be explained later. 26. Example Illustrating Quick Method of Determining Eco- nomic Size of Conductors. For the purpose of working out a practical example the following assumptions have been made : Total kilowatts to be transmitted, P = 12,000. System, three-phase. Power-factor = 0.8. Distance of transmission = 120 miles. Copper conductors to be used, the cost p being $20 per 100 Ib. Percentage to be taken to cover depreciation and annual interest on cost of copper, a = 12.5. Estimated cost of wasted power per kilowatt-year, pi = $22. The economic voltage drop per mile of single conductor will be, by formula (19), P _Q I 12 ' 5 >< 20 ti r = \).65 Aj 22 = 31.5 volts The transmission voltage as given by formula (23) is: Kilo volts = 5.5 Jl20 + = 86 or, say, 88,000 volts at the receiving end. The current per conductor will be: = Watts A/3 X E X cos e 12,000,000 : A/3 X88,000 X 08 = 98.4 amp. 56 ELECTRIC POWER TRANSMISSION J7t Q 1 C Resistance of conductor per mile = -y = T^-T = 0.32 ohm, / yoiTc and since No. 3-0 B. & S. wire has a resistance of 0.326 ohm per mile, that is the standard size which should be adopted unless a more careful study of the complete scheme should lead to a different decision in regard to the pressure of transmission. Since, for a given amount of power to be transmitted, the current will vary inversely as the pressure, it follows that the resistance per mile of conductor to give the economic voltage drop per mile (31.5 volts in this particular example) will be directly proportional to the pressure at which the power is transmitted. Thus if 110,000 volts were found to be a more economical pressure than 88,000, the ohms per mile of conductor would be - - = 0.4, the nearest standard size being 00 No. 2-0 (ohms per mile = 0.41). Power Lost in Line. If w stands for the total I Z R watts lost in the three conductors, based on the calculated value of the resistance, then w = 3 X length of line X / X E r = 3 X 120 X 98.4 X 31.5 = 1115 k.w. on the assumption that a transmission pressure of 88,000 volts is adopted; and since the total kilowatts transmitted are 12,000, the percentage power loss is : 1115 X 100 12,000 : 9-3 Per cent. Voltage Regulation. The drop in pressure per conductor, due to ohmic resistance only, will be: E r X length of line = 31.5 X 120 = 3780 volts or 3780 X \/3 = 6550 volts between wires, since the system is three-phase and the volts E r refer to a single conductor only. The percentage ohmic drop is, therefore: 6550 OQ = 7.44 per cent. oo This figure alone does not, however, give much indication as to what will be the actual regulation of the line, as the effects of inductance and electrostatic capacity must be taken into account and the resultant difference of pressure between the transmitting and receiving ends of the line calculated by any ECONOMIC PRINCIPLES AND CALCULATIONS 57 one of the usual methods. The resultant pressure drop may be found to be excessive; it may be such as cannot readily be dealt with in a practical scheme, and in such a case the economy of the line may have to be sacrificed by putting in larger conductors. It is obvious that other conditions may render it inexpedient or impossible to adopt the most economical size of conductor as calculated by the application of Kelvin's law, but in such cases experience and common sense will usually indicate the proper course to follow. If the economic size of wire is small it is possible, but not probable, that there may be trouble due to excessive heating. A want of mechanical strength, or loss of power due to corona formation, are more likely to lead to the selection of a conductor diameter larger than the "economic size." If, on the other hand, the conductor diameter is very large, there may be difficulties in handling and in taking the strain on the individual insulators. The remedy in this case is obviously to subdivide the single circuit into two or more parallel circuits, and, in fact, there are many advantages in doing so rather than running very heavy single conductors. One particular aspect of the question of subdivision of transmission lines is dealt with in Appendix I. Again, even from the economic point of view, the case might arise of a temporary installation intended to give a quick return on capital invested, and an exceptionally small size of wire giv- ing a large I Z R loss might produce the best results. This, how- ever, leads to the consideration of the most important factor in the whole problem, namely, the correctness of the estimates of costs, depreciation allowances, and power transmitted, upon which the value of the calculated results will mainly depend. It is here that the experience, foresight and sound judgment of the engineer must necessarily play an important part, and it is not possible in this chapter to do more than draw attention to some considerations which must not be overlooked. 27. Estimation of Amount and Cost of Energy Wasted in Conductors. The correct value of the power (P) from which the value of the current (/) is determined is frequently very difficult to estimate. This is a point which is best considered when determining the cost of the wasted energy. It is, however, clear that the annual amount of energy wasted will depend not only on the average value of 7 2 , but also on the time during 58 ELECTRIC POWER TRANSMISSION which the average amount of power may be considered as being transmitted by the wires. If, therefore, it is desired to estimate accurately the amount of energy wasted annually in the lines, a probable load curve for the year should be drawn and the aver- age I z calculated therefrom. This will give a value for I which, if considered as flowing in the wires continuously throughout the year, will lead to a certain watt-hour or yearly energy loss, the cost of which it is desired to know. Now, the annual cost of production of an additional elec- trical horse-power, considered apart from the total cost of pro- duction, is always 'difficult, if not impossible, to estimate ac- curately, but where coal is the source of energy there is at least the extra cost of coal consumed to be taken into account when estimating the production cost of the lost energy. The case is different in a water-power generating station, where the cost of running the station at full output is very little in excess of the cost of running at one-quarter or one-tenth of maximum output, and it is even more difficult to decide upon a figure which shall represent the cost of wasted energy (pi in the cal- culations) with sufficient accuracy to make the calculations of the economic conductor of real practical value. There are two points in connection with water-power proposi- tions which must never be lost sight of: (1) If the amount of water-power available is limited, while the demand for power is unlimited, the cost (pi), of the wasted energy may be taken at the price which the user would actually be prepared to pay for it were it available for useful purposes. (2) If the water-power is unlimited as compared with the de- mand for power, the cost of wasted energy is practically nil, ex- cept for the fact that a generating plant has to be installed of a somewhat larger capacity than would otherwise be necessary; and the works cost of the wasted energy must, of course, include a reasonable percentage to cover interest and depreciation on this extra plant. 28. Estimation of Percentage to Cover Annual Interest and Depreciation on Conductors. So far as interest is concerned, if cash is to be paid for the conductors, the figure to he taken for interest on capital should be on a par with the expected per- centage profit on the complete undertaking; but if the conductors are mortgaged, it is the annual amount of the mortgage which should be taken. ECONOMIC PRINCIPLES AND CALCULATIONS 59 In regard to depreciation, the probable life of the con- ductor must be estimated, and this, to a certain extent, may depend upon the life of the transmission line considered as a whole. 29. Economic Voltage. It should be clearly understood that the foregoing articles deal only with the determination of the correct size of Conductors based on certain assumptions as regards voltage and power to be transmitted. The cost of generating and transforming plant and buildings, as influenced by the volt- age, must be carefully considered, together with the type and cost of pole line, so far as these are influenced by the size of the con- ductors. The character of the country, too, will have some bear- ing on the design of the transmission line, and the final choice of voltage may depend to some extent upon whether a wood pole line with comparatively short spans and (preferably) small spacing between wires is likely to be more economical than a line with steel towers which will permit of longer spans with wider spacing between wires. In other words, the total cost of the whole undertaking and the total annual losses of energy from all sources, as influenced by any change of voltage, must be con- sidered before the line pressure as given by formula (23) can be definitely adopted as being the most economical for the under- taking considered as a whole. Clearly, the choice of the transmission voltage is a very im- portant matter; and since it is possible to determine the proper voltage on purely economic grounds, the use of exceptionally high pressures merely because of theh 1 interest from an engineering standpoint, should be discouraged. On the other hand, it would appear that most transmission-line troubles occur on lines work- ing at pressures between 30,000 and 80,000 volts; and an im- portant consideration to bear in mind is that more trouble may be experienced with heavy currents than with high voltages, owing to the more serious effects of interruptions or transient disturbances when the current is large, so that greater security may sometimes be obtained by increasing the voltage with a view to reducing maintenance and operating costs. When figuring on the best voltage for any particular scheme, the capital cost of all works, buildings, or apparatus, which is liable to be influenced by the transmission-line pressure, together with all operating and maintenance charges which may be simi- larly influenced, must be taken into account. It will usually 60 ELECTRIC POWER TRANSMISSION be found convenient to reduce all such costs or differences of cost to the basis of annual charges. 30. Costs Other Than Transmission Line, Liable to be Influ- enced by Voltage Variations. The cost of a generating station complete with all plant and machinery, but not including trans- mission line, may be anything from $25 to $200 per kilowatt installed. It will depend on total output, that is, on the size of the station, on location, and transport and labor facilities. The cost of a hydro-electric station will depend on the head of water, the amount of rock excavation, the size of dam, length of tunnels and penstocks, etc. The figures given in the accompanying table are approximate costs per kilowatt (not including the transmission line) for a medium-head hydro-electric development suitable for a total output in the neighborhood of 10,000 k.w. to be transmitted over Transm ission-line voltage 30,000 60,000 100,000 Hydraulic works outside power-station buildings. . Power-station building, including excavations Receiving-station building Switch-gear (both ends) Electrolytic lightning arresters $15.00 5.00 1.00 1.20 0.34 $15.00 5.06 1.03 1.35 0.66 $15.00 5.10 1.05 1.70 1.20 Transformers (both ends) 2.50 2.90 3.50 Generators and exciters 8.00 8.00 8.00 Cables in buildings entering bushings etc 40 40 0.50 Crane, sundries, and accessories, including pre- liminary work 2.00 2.10 2.20 Turbines and hydraulic equipment 10.00 10.00 10.00 Total cost per kilowatt $45.44 $46.50 $48.25 two outgoing three-phase feeders. The usefulness of these figures lies mainly in the indication they give of the probable differences in cost with the variation of transmission-line pressure. 31. Annual Charges Depending on Voltage. These charges may be summarized as follows: 1. A percentage on all capital expenditure, whether for gener- ating station, transmission line, or receiving stations, which is not constant irrespective of voltage. 2. The yearly cost of the power lost in the transmission line. ECONOMIC PRINCIPLES AND CALCULATIONS 61 3. The yearly cost of power lost in generators and transformers (the efficiency of the electrical plant will not necessarily be the same for all voltages). 4. The yearly cost of maintenance and operation. This may depend upon length of spans in transmission line, and on the necessary plant, switch-gear, etc., to be attended to, and kept in working order. The percentages referred to under item (1) must include in- terest on capital invested and depreciation. 32. Depreciation. Depreciation is the loss of value or com- mercial utility due to deterioration with age. The term may be used to cover loss of value resulting from very different causes. A distinction should be made between natural and functional depreciation. Natural or physical depreciation is the loss of value due to physical or chemical changes which, in time, will render the machine or plant practically, useless. Atmospheric changes, alternations of heat and cold, wear and tear, erosion, rust, decay, electrolysis, are causes of natural depreciation. Functional depreciation is the loss of value due to the fact that, with the lapse of time, the machine, plant, or structure under consideration does not function as efficiently as when it was first put into use, or as efficiently as it should function to compete with improved methods or apparatus. It may be- come inadequate owing to rapid growth in the demand for the service which it is intended to render; or it may become obsolete. Thus, functional depreciation may be due to either inadequacy or to obsolescence. A machine or structure becomes obsolete owing to scientific or artistic developments, i.e., inventions. It is practically impossible to predict a future state of development in any branch of engineering, and the proper amount to allow for depreciation is largely a matter of guesswork based upon previous experience. A sinking fund for the creation of a depreciation reserve should be formed by placing annually at compound in- terest a certain sum of money which, at the end of the estimated life of the structure or plant, will reproduce the sum originally invested. The accompanying table has been worked out on a basis of 5 per cent, compound interest. It gives the amount, in dollars, which must be put aside each year in order to provide a fund of $100 at the end of a term of years after which the value of the works or materials under consideration is assumed to be nil. 62 ELECTRIC POWER TRANSMISSION DEPRECIATION TABLE (On basis of 5 per cent, compound interest earned by money put aside annually) Life, years Depreciation, per cent. Life, years Depreciation, per cent. 2 48 70 28. 1 710 4 23 20 30 1 505 6 8 14.70 10.50 32 34 1.325 1.175 10 7 95 36 1 045 12 6 28 38 928 14 5.10 40 0.828 16 18 20 22 4.23 3.55 3.03 2 60 42 44 46 48 0.740 0.662 0.593 532 24 2 25 50 477 26 1 96 Although it is rarely necessary to consider scrap values, an exception should be made in the case of the copper conductors of transmission lines. If D is the price originally paid for the material, and S is the estimated scrap value at the end of n years, the percentage of the original sum D to be put aside annually to cover depreciation is not r per cent. as calculated for zero value at the end of n years but r f per cent, of which the value is The "life," in years, of any part of a machine or structure is very difficult to estimate. It is here that the distinction between natural and functional depreciation becomes important, because whichever one appears to indicate the shortest life, should be considered to the exclusion of the other. Thus, the life of wood poles liable to decay and attacks by insects will lead to the allowance for natural depreciation being larger than for func- tional depreciation, and the latter can therefore be ignored. But there are many kinds of plants, such as generators of inefficient design or insufficient capacity, of which the life determined on a basis of functional depreciation is shorter than their probable ECONOMIC PRINCIPLES AND CALCULATIONS 63 wearing possibilities, and it is then the natural depreciation which should be ignored. 33. Example: Method of Determining Most Economical Voltage. Consider the case of a typical medium-head hydro- electric power station: Distance of transmission = 50 miles. Duplicate three-phase line with copper conductors. Cost of copper conductors = $20 per 100 Ib. Power demanded = 15,000 hp. or 11,200 k.w. (It is assumed that this power will be required continuously day and night for industrial purposes, and that it is the probable limit of the water- power available.) Power factor = 0.8. Selling price of power = $21 per horsepower-year. Interest on capital invested; allow 6 per cent. The economic drop of voltage per mile of single conductor as given by formula (19) is: Er = g. 35 J

^ d CO (N ^H ^H ^H - - - -2 - v v w o ic ^ tf" >* SB Q "3 ^ ^(Ni-HGO ^OO oOC<>CO CO "3 COCOON OOO * '' I 3 : e .15 - 1 :^x.2^ jii-li!- II ii I ill^Ml irl 5 -5 w ^> "S e J , > > lil^ll .2r TS -d 13 sS I 1 I I I 1 o .9 II | fl s^ 5 Si -3 S-S | s^ .2 S28& 5 -^ rti ill i| $ I I s -S.g il 76 ELECTRIC POWER TRANSMISSION a temperature of 60 F. The sizes of the smaller conductors are given in the B. & S. gauge because this is generally used on this continent. With this system, when the area of any particular gauge number is known, it is only necessary to double this in order to get the area of the third size larger; or if instead of multi- plying by two, the multiplier 1.261 is used, this will give the area of the next size larger in the B. & S. series. It is convenient to remember that No. 10 B. & S. copper wire measures almost exactly ^f o m - m diameter, and has a resistance of 1 ohm per 1000 ft. When the resistance, R, per mile of a stranded conductor is known, the weight per mile is approximately: For Copper; pounds per mile= -5- 440 For Aluminum; pounds per mile = ~rr RESISTANCE AND WEIGHT OP STRANDED CONDUCTORS Size, cir. mils and B. & S. gauge Diameter, inches, approx. Circular mils, nominal Area, sq. in., approx. Copper Aluminum Ohms per mile Weight per m,le, Ohms 3L Weight per mile, Ib. 600,000 0.89 600,000 0.472 0.0920 9750 0.153 2920 500,000 0.81 500,000 0.393 0.1095 8100 0.182 2430 450,000 0.77 450,000 0.354 0.1210 7300 0.202 2187 400,000 0.73 400,000 0.314 0.1363 6500 0.227 1944 350,000 0.68 350,000 0.275 0.1566 5650 0.260 1701 300,000 0.63 300,000 0.236 0.1818 4880 0.303 1458 250,000 0.58 250,000 0.1965 0.2192 4060 0.364 1215 4/0 0.53 211,600 0.1661 0.260 3448 0.430 1028 3/0 0.47 167,800 0.1317 0.326 2730 0.542 816 2/0 0.42 133,100 0.1045 0.410 2165 0.684 647 0.37 105,600 0.0830 0.518 1705 0.862 513 1 0.33 83,700 0.0657 0.655 1346 1.085 407 2 0.29 66,400 0.0521 0.826 1067 1.370 323 3 0.26 52,600 0.0413 1.040 850 1.728 256 4 0.23 41,700 0.0327 1.313 675 2.185 203 5 0.207 33,090 0.0260 1.635 540 2.720 162 6 0.183 26,250 0.0206 2.091 422 3.470 126 ELECTRICAL PRINCIPLES AND CALC ULA TIONS 77 42. Skin Effect. Imagine a straight length of cable of fairly large cross-section, through which a steady continuous current is flowing, the return circuit being a considerable distance away. The magnetic induction due to this current will not be only in the non-conducting medium surrounding the wire, but a certain amount due to the current in the central portions of the cable will be in the substance of the conductor itself. In other words, the magnetic flux surrounding one of the central strands of the cable will be greater than that which surrounds a strand of equal length situated near the surface. It follows that, if the circuit be now broken, the current will die away more quickly near the surface of the conductor than at the center; and, for the same reason, on again closing the circuit, the current will spread from the surface inward. If, now, the conductor be supposed to convey an alternating current, it is evident that, with a sufficiently high frequency (or even with a low frequency if the conductor be of large cross- section), the current will not have time to penetrate to the in- terior, but will reside chiefly near the surface. This crowding of the current toward the outside portions of the conductor has the effect of apparently increasing the resistance; and it follows that if I is the total current in a cable of ohmic resistance R, the power lost in watts would no longer be I 2 R, as in the case of a steady current, but I Z R', where R' which stands for the apparent resistance of the conductor is k times greater than R, its true resistance. The multiplier k may be read off the diagram Fig. 20, or if preferred, it can be calculated by means of the formula: (24) where F is a factor proportional to the vertical distances on the diagram, that is to say, to the quantity area of cross-section X frequency. The value of F for copper is : F = 0.0105d 2 / and for aluminum, F = 0.0063d 2 / where d is the diameter of the conductor in inches, and / is the frequency in periods per second. This formula and the curves of Fig. 20 are based on the assumption that the return current is at an infinite distance; but this assumption introduces 78 ELECTRIC POWER TRANSMISSION no appreciable error when dealing with overhead transmission lines. It will be observed that, so long as the product d z f remains unaltered, the multiplier k is constant provided the material ! 1.00 L02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18 1.20 1.22 1.24 1.26 L28 1.30 "Skin Effect" Multiplier (&) FIG. 20. Diagram giving "skin effect" coefficient. remains the same. Thus if, when doubling the frequency, the sectional area of the (circular) conductor is halved, the resistance to alternating currents ratio resistance to continuous currents remains unaltered. ELECTRICAL PRINCIPLES AND CALC ULA TIONS 79 In regard to the material of the conductor, the value of F in the formula is directly proportional to the specific conduc- tivity of the metal so long as the frequency remains constant. Thus if F (or the value of the ordinates in the diagram, Fig. 20) is known for a conductor of given diameter, made of copper, its value for any other "non-magnetic" material is given by the ratio: conductivity of metal of conductor conductivity of copper If the conductor is of iron (or other "magnetic" material), the value of k may be much greater than this ratio would indicate. This point will be taken up again in Article 44. It is a not uncommon belief that when aluminum conductors are used in place of copper, the larger diameter necessary to give the same conductivity will lead to a greater loss through "skin effect;" but the above multiplying ratio makes it clear that the percentage increase of losses with alternating currents of the same frequency will be independent of the material of the conductor (iron excepted), because the greater sectional area necessary to maintain the same ohmic resistance of the lines when a wire of lower conductivity is used, is evidently exactly balanced by the higher specific resistance of the metal. The increased pressure drop and PR loss on overhead lines at normal frequencies and with conductors of average size are usually very little greater with alternating than with con- tinuous currents; but when the material is iron or steel the difference may be very noticeable, and in such cases as the rail return of an alternating current traction system, it should be taken into account. 43. Inductance of Transmission Lines. For the purpose of calculating the flux of induction outside a straight cylindrical conductor, it is permissible to assume that the current is con- centrated on the center line of the wire. The lines of magnetic induction surrounding a long straight wire carrying an electric current of which the return path is at a considerable distance, will be in the form of circles concentric with the conductor. The number of lines, or flux in maxwells, contained between any two imaginary concentric cylinders, of average radius x centimeters, 80 ELECTRIC POWER TRANSMISSION and axial length I centimeters (see Fig. 21) will be the product of the magnetomotive force by the permeance, or 4r T . Z X dx X*t 2irx 10 dx X x where / is the current in the wire, /* is the permeability, and dx is the separation between the cylinders, in centimeters. Assuming dx to become smaller and smaller without limit, and putting p = 1 (for the condition of flux lines in ah*), the ex- Fia. 21. pression for the total flux outside the conductor, up to a limiting distance of d centimeters, is 211 (25) 211 , fd\ To log f Induced volts = , J (26) If I stands for the virtual value of the alternating current, the maximum value of <, by formula (25), will be 10 Substituting in formula (26) after replacing I by the number of centimeters in a mile, and converting the Napierian logarithms into common logarithms, we get Volts induced per mile of conductor = 0.00466/7 log (-) (27) This formula is approximately correct for conductors of overhead transmission lines when these are of "non-magnetic" material; but it should be slightly modified to take into account the effect of the flux lines within the material of the conductor. This additional drop of pressure is not easily calculated because different amounts of flux link with different portions of the conductor. It is obvious that a portion of the conductor near the surface is surrounded by fewer flux lines than a portion near the center of the cross-section. The result is that the e.m.f. ELECTRICAL PRINCIPLES AND CALCULA TIONS 83 induced per unit length of conductor is not the same throughout the cross-section. This suggests the possibility of eddy currents in the wire; but what actually takes place is a distribution of the current density over the cross-section such that the total impedance drop or the IR drop added (vectorially) to the IX drop will have the same value at all parts of the conductor cross-section. The correct calculation of the internal reactance drop for "non-magnetic" cylindrical conductors is given in Prof. H. B. Dwight's book on Transmission Line Formulas. The result is that the inductive pressure drop is actually somewhat greater than as indicated by formula (27), which neglects the internal flux. The corrected formula is 0.000506 ] (28) The antilogarithm of the constant in the brackets is 1.285, and it is more convenient to write the formula Reactive voltage drop per mUel _ , /A of single conductor J \ rl which is the same as formula (6) already given in Chapter II. The reactance of stranded cables is slightly less than that of solid conductors of the same cross-section, owing to the fact that the overall diameter of the cable is greater than that of the solid wire. Excellent tables giving inductive reactance in ohms per mile for different spacings and sizes of wires are given in the Stand- ard Handbook for Electrical Engineers; these figures, when mul- tiplied by the value of the current flowing in the conductor, give the induced volts as calculated by formula (29). 46. Iron as a Material for Transmission Line Conductors. The European war, by limiting the supply of copper and aluminum available in Germany, and by causing an abnormal increase in the price of these metals all over the world, has led electrical engi- neers to consider the possibility of using other metals as conductors of electricity. Zinc has been used in Germany for insulated wires and cables; but it is mechanically weak, and generally unsuitable for overhead transmission lines. When considering the economic advantages of using iron or steel conductors, it is necessary to take into account: (a) the cost of the material at the place where it is to be used ; (6) the 84 ELECTRIC POWER TRANSMISSION "life" of galvanized iron wires or cables as compared with that of copper and aluminum; (c) the energy losses in trans- mission; (d) the voltage regulation, and the increased cost (if any) of maintaining the pressure within specified limits at the receiving end of the line. Under item (c) the greatly increased "skin effect" with alter- nating currents must be taken into account as well as the higher specific resistance which requires a larger cross-section of iron than of copper wire even when the transmission is by continuous currents. Under item (d) the internal inductance of the wire which is almost negligible with copper or aluminum becomes a matter of considerable importance owing to the greatly increased magnetic flux in the material of the conductor when iron or steel is used. Although cables of extra high strength steel wire are occasion- ally used for long spans such as river crossings on important overhead lines transmitting large amounts of energy, this material would not be satisfactory as a substitute for copper or aluminum except on comparatively short sections of the entire line. It seems, however, that iron or steel conductors may be used to advantage on short-distance small-power transmissions when the price per pound of copper wire has been forced up to 30 cents or more. On account of the wide variations in the electric and magnetic qualities of the different grades of iron and steel wire, it is prac- tically impossible to predetermine losses and pressure regulation with a high degree of accuracy. The particulars and data, together with the numerical example, in the following articles should, however, be helpful to the reader when making prelimi- nary calculations on iron wire transmission lines. 47. Apparent Resistance of Iron and Steel Conductors. The relative resistances of iron and copper wires were given in Article 41, and wire tables will be found in the Handbooks for Electrical Engineers; 1 but the accompanying table includes the sizes likely to be used in practice. The figures give the approximate resistance to continuous currents and must be multi- plied by the skin effect factor when the current is alternating. 1 Very complete particulars relating to conductor materials will be found in the Handbook on Overhead Line Construction published by the National Electric Light Association. ELECTRICAL PRINCIPLES AND CALC ULA TIONS 85 APPROXIMATE RESISTANCE PER MILE OF SOLID GALVANIZED IRON WIRE AT 68 F. Gauge No., B. W. G. Diameter, in. Ohms per mile Weight lb. per mile E. B. B. B. B. 2 0.284 4.1 4.9 1160 3 0.259 4.9 5.8 960 4 0.238 5.8 6.9 810 5 0.220 6.8 8.1 690 6 0.203 8.0 9.5 590 7 0.180 10.2 12.1 460 8 0.165 12.1 14.4 390 9 0.148 15.0 17.9 315 10 0.134 18.2 21.7 260 7-strand Ke-in. galv. steel (ordinary) . 5.4 ohms per mile. . 1110 lb. per mile 7-strand y-\n. galv. steel (ordinary) . . 8.6 ohms per mile. . 660 lb. per mile 7-strand Ke-in- Siemens-Martin steel. 7.4 ohms per mile. . 1110 lb. per mile 7-strand ^-in. Siemens-Martin steel. . 9.6 ohms per mile. . 660 lb. per mile 7-strand K-in. E. E. B. iron 7.8 ohms per mile . . 660 lb. per mile The resistance of ordinary steel wire is about 30 per cent, higher than that of the E. B. B. iron. The skin effect coefficient will depend not only upon the diame- ter of the wire and the frequency, but also upon the resistivity and magnetic properties of the iron or steel. The magnetic permeability will, in its turn, be some function of the current in the wire, and it is not possible to express the skin effect co- efficient (k) by means of a simple formula as was done in Article 42 in connection with "non-magnetic" conductors. The co- efficient k, for a frequency / = 60, as calculated from tests on certain samples of iron and steel conductors, may be obtained from Fig. 23. When the true ohmic resistance, R, of the iron conductor, is multiplied by the skin effect factor (A;), the product, R', will be the effective resistance of the wire to an alternating current of the given frequency (in this case 60 cycles per second). In other words, if the power wasted in heating the wire with con- tinuous currents is PR, it will be 7 2 (kR) when carrying an alternating current of virtual value 7. 48. Internal Reactance of Iron and Steel Conductors. The formula (28) in Article 45 gives the total inductive voltage drop 86 ELECTRIC POWER TRANSMISSION in a mile of "non-magnetic" conductor; the term 0.000506/Z being the pressure drop due to the flux lines within the material of the conductor. Obviously, if the permeability is no longer H = 1, but a larger number, the loss of pressure will be greater, and this is what occurs with iron conductors. It is a very simple matter to write Voltage drop per mile of single conductor 1 _ ft nn nsnA/7 v due to internal reactance / = 2.6 01 23 4567 8 9 10 11 12 13 14 Currents Amperes, FIG. 23. Curves giving skin effect coefficient for iron conductors for a frequency / =60. but in this connection n is a purely imaginary number repre- senting an "equivalent permeability" which cannot be calcu- lated, and which is, in any case, some function of the current (/) and the frequency (/). A formula in this shape is therefore practically worthless, and it is necessary to rely on test data obtained from sizes and grades of wire approximating to those of the conductor it is proposed to use. ELECTRICAL PRINCIPLES AND CALCULATIONS 87 One of the most valuable contributions available for the use of those desiring to calculate the probable regulation and losses in lines using iron or bi-metallic conductors, is the Paper No. 252 by J. M. Miller issued by the Bureau of Standards at Washington, D. C. Additional data will be found in the article by Messrs. C. E. Oakes and W. Eckley published in the Electrical World of Oct. 14, 1916, in the article by L. W. W. Morrow in the Electrical World of July 14, 1917, and in the article by C. E. Oakes and P. .-270 a I i* Approximate Vajnei of Internal Beactance Voltage Drop ( Vi ) In Iron and Steel Conductors carrying Alter- nating Current! at /.= .60. 2 a i 5 6 7 8 9 10 11 12 13 14 15 16 17 FIG. 24. Internal reactive voltage drop in iron conductors. A. B. Sahm in the Electrical World of July 27, 1918. Data from these sources have been used in preparing the curves of Figs. 23 and 24. 1 For calculating the total inductive voltage drop in an iron wire transmission line, a modification of formula (28) may conven- iently be used, because it is desirable to distinguish between the external reactance which depends only upon the size and spacing of the conductors, apart from the material and the internal 1 A valuable collection of data referring to iron wires for transmission lines will be found in Prof. W. T. Ryan's article "Iron wire for short high-voltage lines" in the Electrical Review (Chicago) Sept. 22, 1917, Vol. 71, p. 496. 88 ELECTRIC POWER TRANSMISSION reactance which will be greater with "magnetic " than with "non- magnetic" materials. If L is the external inductance (coefficient of self-induction), in henrys, of the conductor due to the flux of induction outside the wire, and L is the internal inductance due to the flux of induction inside the wire, the total reactance (in ohms) being X (total) = X (external) + X (internal) may, on the sine wave assumption, be written X (total) = 27T/L + 2 7 r/L i and the reactive drop (in volts) when the current is / amperes is IX = 27T//L + 27T//L, The value of L per mile of conductor is 0.000741 log - whence Total reactive drop in volts ] , per mile of single iron or \ = 0.00466// log - -f V t (30) steel conductor J where F< = 2-jrfFLi and has to be determined experimentally. Its value, for a frequency / = 60, may be read off Fig. 24 which, however, refers only to a limited number of sizes and kinds of wire. 49. Example of Calculations for Iron Wire Conductors. Given a transmission line 10 miles long consisting of No. 4 E. B. B. galva- nized iron wires spaced 3 feet apart, carrying a current of 5 amperes at a frequency of 60; calculate (a) the loss of power, and (6) the loss of pressure, in each wire. (a) The D.C. resistance (from wire table) is R = 5.8 X 10 = 58 ohms. The skin effect factor (from Fig. 23) is k = 2.2 whence R' = 58 X 2.2 = 127.5 ohms The watts lost = PR' =5 X 5 X 127.5 = 3.19 k.w. per wire. (6) The internal reactive pressure drop per mile (from Fig. 24) is Vi = 37 volts, whence IX (total) for 10 miles of wire (by formula (30) is 10 X 0.00466 X 60 X 5 X log ^r^ + 370 = (say) 405 volts u. i iy The IR f drop being 5 X 27.5 = 638 volts, it follows that the impedance drop is IZ = V(638) s + (405) 2 = 755 volts ELECTRICAL PRINCIPLES AND CALC ULA TIONS 89 This figure does not, however, necessarily represent the differ- ence in pressure between the generating and receiving ends of the line; but this point will be taken up in the following article. 50. Inherent Regulation of Transmission Line. Regulation Diagrams. The fundamental vector diagram, Fig. 10, which was described in Article 9 of Chapter II, is reproduced here for convenience. The resistance drop CB (of which the numerical value was 638 volts in the foregoing example) is drawn parallel to the current vector, while the reactance drop DC (of which the numerical value was 405 volts in the example) is drawn at right angles to the current vector. The impedance drop is DB, (I) A Fia. 10. Vector diagram for line calculations capacity neglected. but this does not correspond to the loss of pressure in trans- mission except when the angle happens to have the same value as the angle 9. The difference in pressure between the genera- ting end and receiving end voltages maybe calculated as explained in Article 9, and it will depend not only on the resistance and size and spacing of the conductors, but also on the power factor of the load, since this will determine the position of the point B on the dotted circle. It is the position of the point B on the circle that modifies the ratio of the length FD to the length BD, even if the proportions of the impedence triangle BCD remain unaltered. Problems can be solved graphically by drawing the diagram, Fig. 10, to the proper scale; but the objection to this method is that the radius OB is generally large in proportion 90 ELECTRIC POWER TRANSMISSION to the quantities represented by the impedance triangle, and the process is either tedious or the results are unsatisfactory. The field for ingenuity in the construction of practical charts based .1 Power Factor 1.0 6* 10,? 15? 20? 25? 30* 35% Curved Lines- -Percentage Voltage Loss FIG. 25. Mershon diagram for determining voltage regulation. on the fundamental diagram (Fig. 10) is very great. One of the methods of obtaining graphical solutions is with the aid of the Mershon diagram. ELECTRICAL PRINCIPLES AND CALCULATIONS 91 In Fig. 25 curves concentric with the dotted circles of Fig. 10 are drawn on a piece of squared paper from a center which lies on the prolongation of the base line, but at a considerable dis- tance outside the diagram. The radius of the inner circle is 10 (or 100) divisions in length, and the projection on the horizontal axis of any point B is therefore the cosine of the angle BOA of Fig. 10 and it indicates directly the power factor at the receiving end. By expressing the calculated resistance and reactive voltage drops as percentages of the receiving end pressure, the impedance triangle can readily be drawn to the proper scale, and by making the space between the circles equal to the side of the squares on the divided paper, the regulation, or difference FIG. 26. Vector diagram illustrating approximate method of determining regulation. between generating and receiving end pressures (FD hi Fig. 10), can be read off the diagram as a percentage of the receiving end pressure. As an example of the use of the diagram, suppose the power factor of the load is 0.77, and that the calculated components of the pressure drop are, Resistance volts = 17 per cent, of receiving end pressure. Reactance volts = 22 per cent, of receiving end pressure. From the division on the horizontal axis corresponding to power factor 0.77 follow the vertical ordinate until it meets the inner circle at B; then measure horizontally 17 divisions, and vertically 22 divisions, and the point D which lies on the dotted 92 ELECTRIC POWER TRANSMISSION circle 27.5 divisions larger in radius than the inner circle (which is described with a radius equal to 100 divisions) indicates that the difference in pressure between generating and receiving ends of the line is 27.5 per cent, of the receiving end pressure. Consider now Fig. 26, which is merely a repetition of the fundamental diagram, Fig. 10, with the addition of a few lines. Drop the perpendicular DM on the radius OB extended beyond the point B. It will be seen that when the angle DOB is small, that is to say, when there is little difference between the power factors at the receiving and generating ends of the line, the dis- tance M N will be very small, and for nearly all practical pur- poses the voltage regulation may be expressed by the ratio 0- instead of ^g> this last being theoretically correct and as given by the Mershon diagram. By adopting the alternative construction, and replacing the arc DN by a straight line per- pendicular to either OD or OB, the necessity for drawing circles from a center outside the limits of a practical diagram is avoided. The method used by Professor L. A. Herdt for the calculation of transmission lines (originally described in the Electrical World of Jan. 2, 1909) employs this approximation; and it is also employed in the method about to be described, which the writer has found very quick and convenient for practical calculations. It will be observed that if the impedance triangle BCD (Fig. 26) be moved round on the point B through an angle 0, so that the hypotenuse BD now occupies the position BDi, the perpen- dicular dropped from DI on the extension to the horizontal line BC, meets this line at the point MI, the distance BM\ being obviously equal to BM. Thus, by revolving the impedance triangle through an angle 6 such that cos 6 = the power factor of the load, the projection of the hypotenuse on any line parallel to the current vector will be a measure of the volts lost in transmission. To apply this method in practice, nothing more is required than a piece of squared paper and a piece of tracing paper. The squared paper is divided into any convenient number of equal parts to represent, horizontally, the percentage ohmic drop of voltage, and, vertically, the percentage reactive voltage drop, as indicated in Fig. 27. On the vertical axis on the left-hand side of the diagram, a power factor scale is provided. This is merely an arbitrary length divided into ten equal parts with suitable sub- ELECTRICAL PRINCIPLES AND CALCULATIONS 93 divisions so chosen as to make use of the horizontal ruling of the squared paper. This scale is used for turning the hypotenuse of the impedence triangle through the proper angle, as will be ex- plained shortly. The method of using the diagram is best explained by working p ( - 25 23 21 19 "I J 13 a llj S g 5 3 1 D <> - > - 0.9 0.8 0.7 0.6 0.5 <* - - - - __ _^- - - -. - ( - -- D, - - Pe rcent ige C hnric Volta ere Dr >p. i and: iesul ant) - n 3 6 7 9 11 13 15 17 19 21 23 25 27 29 """ "< . . -.. <. - . FIG. 27. Author's diagram for determining voltage regulation. out an example. The data used for illustrating the Mershon chart will be suitable: Power factor of load = cos 6 = 0.77 Ohmic volts =17 per cent. Reactive volts = 22 per cent. 94 ELECTRIC POWER TRANSMISSION Place a piece of tracing paper over the diagram (Fig. 27) and mark upon it the point D, 17 divisions to the right of the vertical axis, and 22 divisions above the horizontal axis. Then, with a pin or pencil point held at the point 0, move the tracing paper through an angle of 39 degrees 40 minutes (cos 39 40' = 0.77), bringing the point D to D\, and read on the horizontal axis the distance 27.1, which is the difference between the pressures at the two ends of the line, expressed as a percentage of the receiving- end pressure. The result, as read off the Mershon diagram was 27.5, which might at first sight be thought to be more nearly correct; but, as a matter of fact, the writer's method will usually give more accurate results notwithstanding that the solution is not theoretically correct. This is because the impedance triangle is very much larger for the same size of chart than in the Mer- shon diagram; and the subdivisions are more easily read. When the point Di falls between the two inclined dotted lines drawn on the diagram (Fig. 27), this is an indication that the error introduced by substituting the chord for the arc is less than half of 1 per cent. The use of the power factor scale will now be explained. It is not necessary, as suggested in working out the example, to calculate the angle from the value of the power factor and then set out this angle on the diagram. If in addition to marking the point D on the tracing paper, the position of the point P is also marked, it is merely necessary to move the tracing paper round (on the center 0) until the point P falls on the horizontal line rep- resenting the required power factor, as this will ensure that the point D has been moved through the proper angle. The reason of this will be obvious to anyone possessing even the most ele- mentary knowledge of trigonometry. If it is preferred to work with trigonometrical tables, the formulas (7) to (11) of Article 9, Chapter II may be used instead of the diagram Fig. 25 or 27. 51. Pressure Available at Intermediate Points on a Transmis- sion Line. Referring again to Fig. 10, the volts per phase at gen- erating end are V n and at receiving end, E n , the total drop being (V n E n } volts. It does not follow, however, that the pressure available at a point half way along the line will be V n ( ^ ~) because the power factor is rarely the same at all points. The method of calculating the pressure available at some inter- ELECTRICAL PRINCIPLES AND CALCULATIONS 95 mediate point I/ miles from the supply station on a line of total length L miles, when the effects of capacity are negligible, ,T /. is illustrated in Fig. 28 where C'C = BC (-~\ and OD OD' is a measure of the voltage drop between the supply end and the point considered. The power factor angle at this point will be ^ which can be calculated by making the required changes in the formulas of Article 9. Thus, formula (10) would be written BC' + E n cos OD' = cos \l/ FIG. 28. Vector diagram showing pressure at intermediate point on trans- mission line. and the procedure throughout is exactly the same as if calculat- ing the required generating end voltage on a line (L L') miles in length, to give E n volts at the receiving end when de- livering 7 amperes at a power factor cos 0. Such calculations are usually unnecessary refinements, and the error introduced by assuming a uniform drop of pressure along the line is rarely of any practical importance. The manner in which the values of pressure drop, as calculated for a single conductor, are used in determining the inherent regulation of three-phase lines was explained in Article 11 of Chapter II. The principles governing the inductive effects with any number and arrangement of parallel conductors, are discussed in Appendix I. 96 ELECTRIC POWER TRANSMISSION 52. Capacity of Transmission Lines. The formula given in Chapter II (Article 10) for the capacity in microfarads per mile of conductor, as measured between wire and neutral was: C. - (12) This formula is not theoretically correct and would not be applicable if the distance d were very small in relation to the diameter of the wire (or the radius r); but for overhead trans- missions it is a serviceable formula, and, in the writer's opinion, it may be used in all practical calculations. The question 'of capacity on overhead lines is, however, one of very great impor- tance, especially in view of the increasing pressures and distances of transmission ; and it is felt that some space should be devoted to it, even if it be only to sum up our present knowledge on this subject, and refer the reader to sources from which he can obtain more complete information. The exact formula, 1 which gives the linear capacity in micro- farads per mile between two cylindrical parallel wires is c - 0194 , 3n log (a + V^^l] where a = ~- ; but it is more generally useful to consider the capacity as being measured between one wire and the neutral potential surface. This will be twice the value of the capacity as measured between the two wires ; but, when calculating the charg- ing current, it is the voltage between wire and neutral surface that must be taken, if this latter value of the capacity is used. The formula (31) may be put into another form which is very convenient if tables of hyperbolic functions are available. In the formula (31) common logarithms are referred to in the denominator; but by making the proper correction to the numer- ator and substituting Napierian logs, the denominator becomes log, (a + \/ 2 1) which is the quantity of which the hyperbolic cosine is a. Thus, the inverse hyperbolic cosine of a, or cosh -1 a, is the equivalent of log e (a + V 2 1); and with the corrected numerator, the formula (31) becomes, C - - 447 (32) Lm ~ cosh- 1 a 1 H. Fender and H. S. Osborne in Electrical World, Sept. 22, 1910, p. 667. ELECTRICAL PRINCIPLES AND CALCULATIONS 97 If the capacity per mile of single conductor, measured between wire and neutral, is required, the numerators of these formulas must be doubled, and the correct formula may be written either 0.0388 log (a + Va 2 - 1) or, 0.0895 C/ jn * ^ cosh" 1 a (34) Some excellent practical diagrams based on these formulas are to be found in an article by Dr. A. E. Kennelly which ap- peared in the Electrical World of Oct. 27, 1910. The approximate formula (12) given in Chapter II may be written = 0.0388 m ~ log 2a and, by comparing this with the correct formula (33), it will be seen that the first gives results slightly smaller than the true values; but when a is large, that is to say, when the distance be- tween wires is many times the diameter, the error is negligible. The error only becomes appreciable if a is less than 10, and even if a is as small as 4 (a quite impossible state of things on an overhead transmission with bare wires), the error would be only 0.8 per cent. 53. Capacity of Three-phase Lines. The formulas in the last article give the capacity between two parallel wires as measured from wire to neutral, and in the case of a single-phase transmis- sion, the capacity between the two wires would, as it were, consist of two such capacities in series, and would therefore measure half the value given by these formulas, all as previously men- tioned. It should, however, be noted that it makes no difference which value of the capacity is taken for the purpose of calculating the charging current, provided proper attention is paid to the potential difference available for charging the condenser. In the case of the single-phase transmission, the pressure available for charging the two imaginary condensers in series, is exactly twice the pressure between one wire and neutral. Consider, now, a three-phase transmission with the conductors occupying the vertices of an equilateral triangle, as indicated in 98 ELECTRIC POWER TRANSMISSION Fig. 29. If the radius r of the conductors and the distance d between them are the same as in the case of a single-phase transmission, then the capacity as measured between the wire and neutral is the same for the three-phase as for the single-phase transmission; but the charging current is different because the potential difference across each imaginary condenser is no longer Tjl Tjl but /=> where E stands for the voltage between wires. By 2 v3 treating the three-phase system or indeed any polyphase system as a combination of several single-phase systems each having a condenser connected between conductor and ground, FIG. 29. Distribution of capacity. Three-phase transmission. the calculation of capacity currents becomes a comparatively simple matter, unless great refinements and scientific accuracy are aimed at. It has been shown by Mr. Frank F. Fowle 1 and other careful investigators in this field, that the presence of the conducting ground or other neighboring circuits affects only very slightly the capacity between the conductors of an overhead transmission. By systematic transposition of wires on a long transmission, so that each conductor occupies the same position relatively to ground and neighboring parallel wires over the same portion of the total distance, even these slight unbalancings of the charging currents can be corrected if desired. The electrostatic capacity of underground cables in which the conductors are not only very close together, but are separated by 1 " The Calculation of Capacity Coefficients for Parallel Suspended Wires," Electrical World, Aug. 12, 19, and 26, 1911. ELECTRICAL PRINCIPLES AND CALCULA TIONS 99 insulating materials of which the dielectric constant is no longer unity as in the case of air will be considered in Chapter VII. 54. Charging Current Due to Capacity of Transmission Lines. Although on short, low-voltage, lines the charging current (or capacity current) is so small as to be negligible, this current becomes a matter of considerable importance on long-distance high-voltage transmission lines. In Article 10 of Chapter II the charging current as calculated for an existing overhead transmission amounted to 42.4 amperes, representing an output of 7350 apparent kilowatts from the generating station with the line entirely disconnected from the load at the receiving end. Assuming a sine wave of impressed e.m.f., it is easy to calcu- late the charging current of a condenser of known capacity. The fundamental law of the dielectric circuit is * = #(max.) X C (35) where ^ is the maximum value of the dielectric flux expressed in coulombs; #( max .) is the maximum value of the alternating voltage; and C is the capacity (or permittance) of the condenser, expressed in farads. The charge, or quantity, of electricity i.e., the dielectric flux will reach its maximum value (^) at the instant when the charging current is changing its direction, that is to say, when the current is zero, and since quantity of electricity = current X time, we may write SF = average value of charging current (in amperes) during one quarter period X time (in seconds) of one quarter period * IcX ^ where I c stands for the virtual or r.m.s. value of the charging current, on the sine wave assumption. Let E stand for the vir- tual value of the voltage across the condenser of capacity C farads, then $( max .) \/2 E, and formula (35) becomes whence I c = ZwfEC (36) which is the well-known formula for calculating capacity current when sinusoidal wave shapes are assumed. This is the same as formula (13) of Chapter II, which was given without proof. 100 ELECTRIC POWER TRANSMISSION It is possible to express the charging current on overhead lines in terms of the external inductance, or of the external reactive voltage drop. This is due to the fact that there is a constant relation between the inductance and the capacity, which is independent of the size and spacing of the conductors. Thus, the formula for capacity in microfarads per mile (page 29) is (12) while the formula for the external inductance per mile (page 88) is L = 0.000741 log ^ giving a constant product CmL = 34700 (37) The external reactive pressure drop is (IX) = 27T/L whence a value for L in terms of reactive drop is obtained. This value, substituted in (37) gives Cm = 34,700 (IX) By putting this value of C m in formula (36), we get, 34,700 (IX) X 10 6 W17 (39) 8.78 (IX) X 10- These formulas for calculating the magnitude of the charging current, when multiplied by the length of the line in miles, will give the charging current entering the line at the generat- ing end. The result is usually smaller than the value obtained by measurement on actual lines. The reason is that the assump- tion of sinusoidal impressed e.m.f. is rarely justified, and the irregularities and peaks in the actual pressure wave may cause an increase of charging current amounting to 20 or even 40 per cent, of the calculated value. These considerations emphasize the absurdity of devoting a considerable amount of time to ELECTRICAL PRINCIPLES AND CALCULA TIONS 101 mathematical refinements, or of using complicated formulas of which the increased accuracy is of no practical value seeing that they are based upon assumptions that are never realized. 55. Effect of Distributed Capacity and Inductance. A long transmission line of resistance R ohms, reactance X ohms, and capacity C farads, may be thought of as consisting of n sections, of resistance R/n ohms and reactance X/n ohms, with a condenser of capacity C/n farads shunting the wires at the end of each section. The charging current will fall off in amount as the distance from the generating end increases, and the total current in the conductor being the (vectorial) sum of the charging current and load current will be different in each section. The problem is further complicated by the fact that the voltage will not necessarily be the same across all the imaginary con- densers, so that the reduction of the charging current component will not even follow a "straight line" law. By dividing a long line into a large number of sections, and calculating the pressure drop and power factor at the end of each section, the voltage drop and power losses of the complete line could be estimated accurately; but the work would be tedious and, indeed, unneces- sary. By imagining the number of sections, n, to become larger and larger without limit, we approach the condition of distrib- uted capacity for which accurate mathematical formulas are available. A great deal of excellent work has been done by Dr. A. E. Kennelly, Dr. Harold Pender, Dr. J. A. Fleming, Prof. H. B. Dwight, and others 1 in the matter of simplifying the exact methods of calculation for long lines in which the effects of ca- pacity are not negligible; but these methods nearly all include the use and knowledge of hyperbolic functions in place of the trigo- nometric tables with which all engineers are familiar. It is true that Prof. Dwight, following the lead of Prof. T. R. Rosebrugh, has evolved a fairly simple means of computing line voltages 2 by substituting the method of convergent series, and using complex quantities, thus dispensing with the necessity for tables or charts of hyperbolic functions of angles; but, for the solution of prac- 1 Harold Pender, Electrical World June 8, 1909. A. E. Kennelly and Harold Pender, Electrical World, Aug. 8, 1914. J. A. Fleming, Jour. Inst. E. E., p. 717, Vol. 52, June 15, 1914. A. E. Kennelly, Jour. Franklin Inst., Sept., 1914. H. B. Dwight, Book "Transmission Line Formulas," D. Van Nostrand Co. 2 Electrical World, Sept. 5, 1914. UMNAKY --TAW TVACHn MANTA BARBARA, 51 102 ELECTRIC POWER TRANSMISSION tical power transmission line problems, all of these refined methods of calculation are unnecessary. From the academic point of view, the exact solution of all engineering problems is attractive and occasionally desirable, and when the mathematical work can be so simplified as to be available for use by the average engineer, there is no objection to his using it. On the other hand if the engineer has at his disposal simpler and shorter methods of working which yield results within the required practical limits of accuracy, he should not be criticised for preferring them. He may base his calculations on an assumed sine wave of e.m.f., on a maximum (sine wave) current of 100 amperes, and on an esti- mated power factor of 0.9; but he would expect his line to give satisfaction with an actual full load current which might be anything between (say) 90 and 110 amperes, and he would consider himself lucky if the actual power factor of the load proved to be within 5 per cent, of the value he had guessed at. It is for such reasons that the practicing engineer whose time is valua- ble, and who has a habit of using factors of safety somewhat freely rarely evinces a fervent interest in mathematical re- finements whereby the (theoretical) accuracy of his results may be increased by a small fraction of 1 per cent. On lines over 50 miles in length, the effects of capacity on voltage regulation may be appreciable, and some practical methods of taking into account the capacity current on long transmission lines will be explained in the following article. 66. Electrical Calculation of Lines with Appreciable Capacity. In Article 10 of Chapter II, an example was worked out showing how the charging current at the supply end of a long transmission line could be calculated, and the effect of this current in modi- fying the fundamental vector diagram was illustrated in Fig. 13. Thus, when we imagine a condenser to be connected across the lines at a point where the load current is / amperes on a power factor cos 6, the conditions on the supply side of the condenser will be a current of /' amperes and a power factor of cos 6', these modified values being calculated as follows. The vector diagram Fig. 30 is the same as Fig. 13 except for a few additions. The charging current can be calculated by means of the formula (36), which is I c = 2irfE n C (36) This component of the total line current is drawn 90 in advance ELECTRICAL PRINCIPLES AND CALCULA TIONS 103 of the vector E n = OB which is the pressure across the condenser terminals. By dropping the perpendicular AN on OB and making AM = I c , we obtain OM = I' the resultant or total current in the line on the supply side of the condenser. In order to calculate /' and the new power factor angle 0', we may write, ON= I cos6 NA= I sin 6 NM= I sin e - I c NM 7 Sil1 * ~ ! tan 6 - Ic I cos e FIG. 30. whence 0' and the other trigonometrical functions such as sin 0' and cos 6' can be read off the slide rule or obtained from tables. The line current is cos e' Using this value of current, the sides I'R and I'X of the line impedance triangle can now be calculated, and the procedure for calculating the line losses and line drop will be as described in Article 9 (where the line was assumed to be without capacity) except that /' and 0' must be substituted for I and in the formulas. 104 ELECTRIC POWER TRANSMISSION It will perhaps simplify matters and prevent confusion if some of these formulas are reproduced here with the necessary changes to make them applicable to Fig. 30. The functions of the angle < are V n PR + E n COS ^ n (42) , ... (43) I'X + E n sin 6' tan * ' /'B + J.^.f (44) From this last formula (44) we obtain tf> and therefore cos (the power factor at the sending end of the line), whence V n of which the value, from (43), is COS 57. Numerical Examples Illustrating Use of Formulas for the Calculation of Power Factor and Voltage Drop. The data for use in the calculations is as foilows: System; three-phase. Line pressure at receiving end, E = 66,000 volts. "Star" voltage, E n = *^= 33,100 volts. v 3 Frequency, / = 60. Load = 3000 k.v.a. Power factor of load, cos = 0.9. Length of line, L = 100 miles. Conductors of No. 3 copper cable (radius r = 0.13). Spacing, 8 feet (d = 96). From wire table, the resistance is found to be R = 1.04 ohms per mile. By formula (29) the reactance per wire is X = 0.00466 X 60 X log ( 1.285 X ^^ = 0.832 ohms per mile. By formula (12) the capacity (wire to neutral) is 0388 C m = j- = 0.0135 microfarads per mile. r The load current is 7 = ^ 000 ' 00 = 26.2 amperes. \/3 X 66,000 ELECTRICAL PRINCIPLES AND CALC ULA TIONS 105 By way of illustration, we shall calculate the line drop by imagining (1) the whole of the line capacity to be concentrated at the center of the line, and (2) one-half of the total capacity to be concentrated at each end of the line. Case (1), capacity concentrated half way between sending and receiving ends (Fig. 31). With the star voltage E n = 38,100 at the receiving end, we will first calculate the voltage E' n at the point where the con- denser is supposed to be connected. We could if desired use one of the charts, Fig. 25 or 27; but it may be advisable to use, through- out these examples, the trigonometrical formulas which were developed from the fundamental vector diagram. By formula (9), (26.2 X 50 X 0.832) + (38,300 X 0.436) (26.2 X 50 X 1.04) + (38,100 X 0.9) whence cos < = 0.896 and sin = 0.444. 1 /' Amps > a I Amps t I Load of 1 Power E 1 Factor 1 I co,0 i L i L ! 2 ! 2 1 FIG. 31. Diagram showing total capacity concentrated at center of line. By formula (10), For the other half of the line we must use the formulas de- veloped from Fig. 30. By formula (36), the charging current is L = 27r X 60 X 39,800 (100 X 0.0135 X 10~ 6 ) = 20.25 amperes. By formula (40), _ (26.2 X 0.444) - 20.25 26.2 X 0.896 = - 0.367 From tables, 0' = 20 9', cos 0' = 0.939 and sin 0' = - 0.344 whence, by (41), the line current in this section is /' - 26.2 = 25 amperes, 106 ELECTRIC POWER TRANSMISSION which is in advance of the pressure across the condenser because of the negative sign resulting from the solution of formula (40). The sides of the impedance triangle BCD (Fig. 30) are I'R = 25 X 50 X 1.04 = 1300 volts and I'X = 25 X 50 X 0.832 = 1040 volts. The procedure is now as for the 50 miles of line already cal- culated. Putting <' for the power factor angle at the sending end of the line (the angle 0' = 20 9' being the power factor angle at the other end of this section), we have by formula (9) or (44) 1040 - (39,800 X 0.344) 1 1300 + (39, - 12,660 800 X 0.939) 38,680 r VT^ f Current -J t| I | G3 Load of Power Factor cot $ BesiBtance R ohms: Beactance X ohms FIG. 32. Diagram showing one half of total capacity concentrated at each end of the line. whence $' = 18 degrees; cos <' = 0.951; and sin ' = 0.309; the fact of sin 0' being negative indicating a leading current. By formula (10) or (45), V n = = 40,700 volts. The pressure drop per phase is 40,700 - 38,100 = 2600 volts, or 6.82 per cent, of the receiving end pressure. The current entering the line at the generator end under full load conditions is I' = 25 amperes, on a leading power factor (cos <') of 0.951. It should be mentioned that no high degree of accuracy is claimed for these or any numerical examples worked out in this book. A 10-inch slide rule is used for all calculations. Case (2), half the total capacity concentrated at each end of the line (Fig. 32). The effect of connecting a condenser of capacity C/2 between each wire and neutral at the receiving end, is to modify the power ELECTRICAL PRINCIPLES AND CALCULATIONS 107 factor of the load, the line calculations being based on an imagi- nary load power factor cos 6' instead of the actual power factor cos 6. The condensers at the sending end have obviously no effect on the line drop or line losses, but they will modify the power factor of the load at generator terminals. The calculations of line drop are exactly as carried out for the supply end of the line in Case (1). The charging current is now about half the value previously calculated, I c = 2T X 60 X 38,100 (50 X 0.0135 X 10~ 6 ) = 9.7 amperes. Referring to Fig. 30, we have, by formula (40), (26.2 X 0.436) - 9.7 26.2 X 0.9 = - 73 whence 0' = 4 10'; cos 0' = 0.997, and sin 6' = 0.073. 9 By formula (41) the line current is 26.2 X n ' Q7 = 23.65 u. v/t/i amperes. The resistance drop is I'R = 23.65 X 100 X 1.04 = 2460 ohms. The reactance drop is I'X = 23.65 X 100 X 832 = 1970 ohms. By formula (44) , 1970 + (38,100 X 0.073) tan * = 2460 + (38,100 X 0.997) = whence cos = 0.994, and sin < = 0.110. Thus, by formula (45), - ,700 volts which is the same as the figure obtained by assuming the whole of the capacity to be concentrated at the center of the line. If it is desired to calculate the current and power factor at the generator terminals, we have merely to repeat the process by putting the new numerical values in the formulas. Thus 407 I c = 9.7 1^- = 10.35 amperes, ool and the formula (40) when re-written, becomes /' sin - Ic tan <' = /' cos (f> (23.65 X 0.110) - 10.35 23.65 X 0.994 108 ELECTRIC POWER TRANSMISSION whence cos 0' = 0.95 (leading), and by formula (41), the line current is - 23.65 p = 24.8 amperes. In this example, the capacity current is relatively large because the maximum load on a line 100 miles long would generally be more than the assumed value of 3000 k.v.a.; but nevertheless, both methods give approximately the same results. Comparing the figures, and bearing in mind that no high degree of accuracy is claimed, we have, Approximation (1) (Fig. 31) Approximation (2) (Fig. 32) Percentage line drop Line current at generating end Power factor at generating end .... 6.82 25 . 00 amperes 0.951 (leading) 6.82 24.80 amperes 0.95 (leading) Either method gives results that are sufficiently accurate for practical purposes, even in the case of long high-voltage lines. 1 58. Distinction Between Regulation and Line Drop. The percentage drop of pressure on a transmission line may be de- fined as the difference between the sending end and receiving end pressures expressed as a percentage of the receiving end pressure. Thus, V E Per cent, pressure drop = ^ X 100 Hi and this is the same as the regulation when the capacity current is so small as to be negligible. When the capacity current is appreciable, it will cause the voltage of the unloaded line to be greater at the receiving than at the generating end, as explained in Article 10, Chapter II (Fig. 11); and since the regulation is defined as the change of pressure at the receiving end when the load is thrown off (the supply voltage remaining constant), the regulation of a long high-voltage transmission line will usually be greater than the pressure drop. 1 The manner in which the total current in a long line of appreciable capacity changes both in magnitude and phase, may be illustrated graphic- ally by means of models or diagrams involving the idea of two planes per- pendicular to each other. The writer has in mind diagrams similar to those used by Prof. D. D. Ewing in the Electrical World of Dec. 29, 1917, Vol. 70, p. 1252. ELECTRICAL PRINCIPLES AND CALCULA TIONS 109 The rise of pressure at the end of a long transmission line is independent of the size and spacing of the wires. It may be calculated approximately as follows. In Fig. 33, the IR drop (CB) due to the charging current I c , may be neglected as it has no appreciable effect on the pressure rise (E n V n ) which we shall therefore consider as being equal to the induced volts DC. By formula (36) the charging current is I c = 27r/# n C f m L X 10- 6 where C m is the capacity in microfarads per mile, and L is the length of the line in miles. The induced volts are, IX = 2irfLI c L Uc) (Vnl Fio. 33. Vector diagram showing pressure rise at end of long unloaded line. which, after substituting the above value of I c , become IX = (2*f) z E n (C m L)L* X 10- 6 but, by formula (37), the product C m L has the constant value 04 TYjTy whence the pressure rise (conductor to neutral) of the unloaded line is IX = (En - V n ) = (2irf) z EnL 2 X 2.88 X 10- 11 = l.!4Enf*L* X 10- 9 volts (46) which assumes sinusoidal wave shapes. This pressure rise, expressed as a percentage of the line voltage is Per cent, pressure rise due to capacity and [ = 1.14/ 2 !/ 2 X 10~ 7 (47) inductance of line 110 ELECTRIC POWER TRANSMISSION The regulation, on the sine wave assumption, is therefore equal to Percentage line drop + l.l^L 2 X 10~ 7 ; but the last term is negligible unless the distance of transmission (L) is great. As an example of the application of formula (47), the per- centage line drop in the numerical problem of the last article was 6.82 and the percentage rise at the end of the unloaded line is approximately 1.14 X (60) 2 X (100) 2 X 10~ 7 = 4.11, whence the regulation is 6.82 + 4.11 = 10.93 per cent. 59. Line Losses. Apart from leakage and corona losses, which will be considered in Chapter V, the watts lost in transmis- sion are the I 2 R losses. On a three-phase line these will be w = 3PRL where R is the resistance per mile of conductor corrected if necessary for skin effect and L is the distance of transmission, in miles. When the current entering the line at the sending end is equal to the current leaving the line at the receiving end, the value to take for / in the above formula is simply the load current; but when the effect of the distributed capacity becomes important as on a long-distance high-voltage transmission the question arises as to what particular value of the current should be used in the calculations for line loss. As an alternative to dividing the line into a large number of sections and calculating the current in each section which would involve a considerable amount of tedious work we can calculate the average value of the square of the current over the whole distance of transmission. This calculation is easily made if the effect of voltage drop is neglected, or, in other words, if the amount of the charging current is supposed to diminish in direct proportion to the distance from the supply end of the line. Thus, if I c is the value of the charging current at the sending end, the value of the charging current component of the total current at a point x miles from the generating end of a line L miles long will be I x = ^ (L - x) (48) Referring, now, to the vector diagram Fig. 34, where 7 is the load current and cos 6 is the power factor of the load, the total line current (7j) at a point x miles from the generating end is the sum, or resultant, of 7 and I x . This resultant can be ex- ELECTRICAL PRINCIPLES AND CALC ULA TIONS 1 1 1 pressed in terms of its "in-phase" and "wattless" components thus: Ii = V(o - / z ) 2 + 6 2 and the average value of the square of this quantity is 6 2 + average value of (a 7 Z ) 2 as x increases from zero to its maximum value L. Average of b =/COB a=Isin FIG. 34. Vector diagram showing total current at any distance from end of line. Adding 6 2 to this quantity, we get for the average value of the square of the line current (/* 2 )average = (& 2 + 2 ) ~ /, + |/c 2 = P - I C I sin e + ^/ t 2 (49) and the watts lost in a three-phase line, neglecting corona and leakage losses, are approximately W = 3RL (/^average (50) If the current waves are not sinusoidal, and if the pressure at the sending end is appreciably higher than the receiving end pressure, the average square of the current will not be quite correctly obtained from formula (49). 112 ELECTRIC POWER TRANSMISSION Example ofLineLoss Calculations. Instead of assuming entirely new conditions, we shall use the data of the numerical example in Article 10 of Chapter II. and calculate the line losses (1) when there is no load at the receiving end, and (2) when the load at the receiving end is 10,000 k.w. with a power factor of 0.9. Assuming the conductors to be No. 00 copper throughout, the resistance per mile of wire (neglecting skin effect) will be 0.41 ohm, and the known quantities are therefore: Load at receiving end = \/3 El cos 6 = 10,000,000 watts. Line voltage, E = 100,000. Load power factor, cos = 0.9. Load current, / = 64.2 amperes. Resistance of conductors, R = 0.41 ohms per mile. Distance of transmission, L = 210 miles. Capacity current at generating end "1 J . . ., > = 63. 6 amperes, (as previously calculated) J (1) By formula (49) the average value of the square of the charging current when I = is (/coverage = j^ 2 = 1345. By formula (50) the total line loss is 3 X 1345 X 0.41 X 210 X 10~ 3 = 348 k.w. This is the true output of the generating station (neglecting corona and leakage losses) when the working voltage is applied to the unloaded line; but the k.v.a. or apparent kilowatt output is -\/3 X 100,000 X 63.6 X 10~ 3 = 11,000 k.v.a. (2) By formula (49) the average value of the square of the line current when the load current is 64.2 amperes on a power factor cos 6 = 0.9 (sin 6 = 0.436) will be (/^average = (64.2) 2 X 1345 - (63.6 X 64.2 X 0.436) = 3680 whence the total line loss at full load is 3 X 3680 X 0.41 X 210 X 10- 3 = 952 k.w. 60. Control of Voltage on Transmission Lines. The pressure drop at the receiving end of a transmission line may be compen- sated for by raising the voltage at the generating end as the load increases. There are obvious disadvantages to such a method of operation, and it is better, if possible, to regulate the voltage at the point, or points, where constant pressure is required. This is ELECTRICAL PRINCIPLES AND CALC ULA TIONS 1 13 especially true of transmission lines on which there are substa- tions or branch lines at intermediate points. The necessary steady voltage at the receiving points may be obtained by installing hand operated or automatically controlled variable-ratio transformers or "boosters," which may be either S FIG. 35. "Boosters" on three-phase system. of the type with movable iron core, or with tappings from the windings taken out to a multiple-contact regulating switch. On a delta-connected three-phase system it is not necessary to provide more than two single-phase regulators as these may be connected up as indicated in Fig. 35. Here P and S represent B' FIG. 36. Vector diagram two "boosters" on three-phase system. respectively the primary and secondary windings of the variable- ratio transformers. That these are capable of raising the voltage equally on all three phases to the extent of the volts induced in the secondary coil S will be clear from an inspection of Fig. 36. In this diagram, AB, BC, and CA are the three vectors repre- senting the pressures e before boosting up: A A' and BB' repre- 114 ELECTRIC POWER TRANSMISSION sent the added volts between the terminals A and A' or B and B' (Fig. 35). These added volts are evidently in phase with the pressures indicated by the vectors CA and BC respectively, because the potential difference at the secondary terminals of a well-designed transformer is always in phase with the primary impressed e.m.f. It is only necessary to complete the triangle CA'B' to see that the two transformers connected up in the manner described will do all that is required in the way of raising the pressure on the three-phase circuit. 61. Effect of Boosting Voltage at Intervals Along a Trans- mission Line. If a long transmission line, insulated for a maxi- mum working pressure of (say) 100,000 volts, can be worked as a 100,000-volt line at all times through its entire length, it will be more efficient than if only a portion of it is working as a 100,000- volt transmission while portions farther from the generating end 100 Amps. 20.000 ,000 \Loss in 1st Section = 500 KW. X Loss in 2nd Section = 500 KW. FIG. 37. Method of maintaining pressure on long line. are working at (say) 80,000 volts. By installing boosters along the line to maintain the pressure at or near the maximum working value, whatever the load may be, economies may frequently be effected. It is true that the energy put into the line at inter- mediate points cannot be cheaper and, indeed, is usually more costly than the energy supplied to the line at the generating end; but the booster system allows of the pressure being kept up all along the line, thus effecting economy; provided always that the losses in the boosters themselves, their maintenance, and the necessary allowances for interest and depreciation, do not counterbalance the saving. As an example, consider a single-phase line conveying a current of 100 amperes at an initial pressure of 20,000 volts. Suppose the drop in pressure in the whole length of line to be as great as 10,000 volts; this will leave only 10,000 volts at the receiving end. The power put into the line at generating end = 2000 k.w. The loss hi the line = 1000 k.w. The power available at receiving end = 1000 k.w. Hence, efficiency of line = 50 per cent. ELECTRICAL PRINCIPLES AND CALCULA TIONS 115 Now imagine a booster to be introduced at a point half-way along the line. This booster may be considered as a suitably insulated alternator of 500 k.w. capacity, capable of generating 100 amperes at 5000 volts, the arrangement being as shown in Fig. 37. The drop in the first section of the line is, as before, 5000 volts; and the drop in the second section is evidently similar namely, 5000 volts which means that the total amount of power dissipated in the line is the same as it was before the booster was introduced. But by providing this booster at the middle point of the line, it has been possible to raise the pressure at this point up to the initial value of 20,000 volts, with the result that 15,000 volts (1500 k.w.) are available at the receiving end. The addi- tional power available for useful purposes has, of course, cost something to produce; but the point to be noted is this: by keep- ing up the pressure, it has been possible to transmit a greater FIG. 38. Transformers connected as "boosters" on transmission line. amount of energy to the receiving end of the line without increas- ing the losses in the conductors. If, for the sake of simplicity, the losses in the booster are neglected, the line efficiency is arrived at thus: Power supplied to the line = 2000 + 500 = 2500 k.w. Power lost in the line = 1000 k.w. Power available at receiving end = 1500 k.w. Line efficiency = ^r^-: = 60 per cent. Boosters may be arranged to take their power from the generat- ing end of the line; that is to say, they may take the form of variable-ratio transformers, with hand or automatic regulation, connected up as indicated in Fig. 38. Transformers so connected will provide the additional volts at the cost of a corresponding loss of current. 62. Control of Power Factor. The advantages of operating alternating machinery and systems on unity power factor, when possible, are so well known that it will not be necessary to dis- cuss the matter here. The line losses alone as shown in Article 116 ELECTRIC POWER TRANSMISSION 6 of Chapter II are inversely proportional to the square of the power factor. Thus, if the total I*R loss in the line were 200 k.w. with a power factor of 0.707 (a by no means impossible figure in practice), this loss would be reduced to 100 k.w. if the same total power could be transmitted at unity power factor. The control of power factor is obtained by balancing any excess of inductive reactance with condensive reactance, or vice versa. With a changing load at the end of a long transmission line, the power factor at any given point on the line is continually changing, even if the power factor of the load remains constant; and the most convenient means of providing the reactance necessary to maintain a constant and improved power factor is to install synchronous motors which can be made to draw Synchronous Booster Controlling Line Prewure Synchronous Motor Controlling Power Factor Load 1 FlG. 39. leading or lagging currents from the line by over- or under- exciting their field magnets. The principles underlying the fact that an alternating current synchronous motor can be made to act either as a rotary con- denser or a rotary reactor have been dealt with by the writer elsewhere, 1 and they are, moreover, so generally understood that we shall assume the power factor to be capable of control by merely providing one or more synchronous motors of sufficient capacity and with the necessary equipment for varying the field current, at the desired point on the transmission line. Bearing in mind that power factor control is quite as impor- tant, if not more important, than voltage control so far as the losses and efficiency of transmission are concerned, the arrange- ment shown diagrammatically in Fig. 39 should be preferable to the arrangements of Figs. 37 and 38. Here B is a synchronous generator connected as a "booster" and provided with field regulation in order to control the voltage. It is direct-coupled to the synchronous motor, M, which is pro- vided with field regulation in order to control the power factor. 1 Polyphase Currents, Whittaker and Co., London. ELECTRICAL PRINCIPLES AND CALC ULA TIONS 117 Before deciding to install synchronous machinery to control the voltage and power factor of a transmission line, it is neces- sary to consider the increased cost due to such machinery over alternative methods of pressure control, and compare this with the capitalized cost of the saving in transmission losses due to the improved power factor. A point which should not be over- looked is the possibility of synchronous machinery falling out of step, and so causing troubles and interruptions to supply which are best avoided by installing no auxiliary apparatus which it is possible to do without. 63. Use of Rotary Reactors to Control the Voltage. Although Fig. 39 shows two synchronous machines, each with a particular function to perform, it is not always necessary to maintain the power factor at a constant value, and if the machine M is of FIG. 40. Vector diagram showing components of voltage which cause pressure drop. sufficient capacity and properly designed, it may be used to main- tain constant voltage without the addition of the second machine (B) connected in series with the line. The reason for this is that by changing the power factor or the phase difference between the line current and e.m.f. the inductance of the line and of such apparatus as transformers connected thereto, can be util- ized to provide the required voltage. This is best explained with the aid of vector diagrams. Fig. 40 is similar to the fundamental regulation diagrams (Figs. 10 and 26) except that the ohmic and reactive voltage components of the total pressure drop are shown separately for the "in-phase" and "wattless" components of the total line current. Thus, if E n , I, and cos 6, stand respectively for the 118 ELECTRIC POWER TRANSMISSION "star" voltage, the line current, and the power factor at the re- ceiving end, the IR and IX drops due to the "in-phase" com- ponent, ON, of the current are CB and HC respectively, while the IR and IX drops due to the "wattless" component, NA, of the current are GH and DG respectively. The pressure neces- sary at the generating end is F B = OD, and the pressure drop is DF = V n E n . If, now, we can by means of overexcited syn- chronous machinery so change the power factor that the point D will fall on the dotted circle of radius OB, we shall obtain the condition of constant voltage, i.e., the same voltage at the generating, as at the receiving, end of the line. FIG. 41. Vector diagram illustrating effect of reactors in maintaining constant voltage. Assuming no change in the load current /, the impedance triangle BCH will also remain unaltered; but, by drawing a leading "wattless" current from the line, the current compo- nent NA can be not only annulled, but actually reversed, thus making the line current lead the receiving end voltage, there- by changing the voltage drop due to the reactive component of the current into a voltage rise. This is shown in Fig. 41 where the excitation of synchronous reactors, connected across the line at the receiving end, has been increased until the leading ELECTRICAL PRINCIPLES AND CALC ULA TIONS 1 19 component of the line current is equal to OK. The resultant current is OM with a leading "wattless" component NM, giv- ing the impedance triangle HGD which throws the point D on the dotted circle and makes V n = E n . This is the principle of constant voltage transmission with the pressure regulation obtained by providing the necessary number of variable-field synchronous motors at suitable points on the transmission line, but mainly at the receiving end where the heavy load is taken off. By this method, the reactance of a long line usually an objectionable feature tending to limit the size of individual conductors is actually necessary to the proper regulation of the voltage. The machines used as "rotary reactors" may be synchronous motors from which mechanical power is obtained, or rotary converters, or again, machines specially designed for no other purpose than to regulate the amount and direction of the "wattless" current, in which case they would be installed in the receiving point substations and would be run "idle." The power factor will not necessarily be unity, but the improvement in the power factor will generally permit of more energy being transmitted along a given line than would otherwise be permissible or economical. The most econom- ical cross section of conductor may be used without regard to pressure drop, because where a drop of 10 to 15 per cent, would be about the upper limit with the older systems of regulation, a drop of 25 per cent, (due mainly to the inductance of the line) can be taken care of by synchronous reactors. One of the most ardent advocates of this system of regulation is Prof. H. B. Dwight, whose book 1 should be consulted by those desiring further information on this subject. The charging current of the line was not referred to in connec- tion with the diagram Fig. 41, but it is evident that it must to some extent be helpful in reducing the necessary size of the synchronous motors, of which the capacity is determined by the amount of "wattless" current that they are able to provide. The regulation of power factor (and incidentally of the voltage) by means of synchronous motors is not applicable to short-dis- tance small-power transmissions, and even on long-distance lines transmitting large amounts of energy, the engineer should be careful to consider the whole problem from the economic point of view: there are a great many factors to be taken into account, 1 "Constant Voltage Transmission," by H. B. Dwight, John Wiley & Sons. 120 ELECTRIC POWER TRANSMISSION among which reliability of service and maintenance costs are not the least important. The space taken up by this discussion of a particular system of control may seem excessive in view of the limitations of this book; but with the improvements hi design of electrical machinery and the increasing magnitude of power transmission schemes, there is a possibility of the system being used extensively in the future. It is advocated not only by the manufacturers of syn- chronous alternating-current machinery, but by engineers who have satisfied themselves that economy and good service can, under favorable conditions, be obtained thereby. A notable instance is the transmission line from the hydroelectric plant at Point du Bois in Canada to the city of Winnipeg where two 6000- k.v.a. synchronous motors with automatic regulation are in- stalled for the sole purpose of regulating the voltage by power factor control; the line pressure at the generating station remaining constant. 64. Power Factor of Load. The power factor of the load is not always easy to estimate; it may consist of induction motors of various sizes, together with lighting circuits, all having different power factors. If several circuits of different power factors are connected in parallel, the joint power factor may be calculated by the formula: Cos e = \ . = (51) J. / //! sin 0! + 7 2 sin 2 + . \ h Ui cos 0i + 7 2 cos 2 + . where /i, 7 2 . . . are the currents taken by the various circuits of power factors cos 0i, cos 62, . . . etc. The formula (51) is easily developed by summing up the "wattless" and "in-phase" components of the various currents separately. The quantity in brackets is thus seen to be tan 6, while the complete formula is derived from the well-known relation 1 cos 2 e 1 + tan 2 6 65. Grounded versus Isolated Transmission Systems. Whether or not it is advisable, on three-phase transmissions, to use the star connection with grounded neutral, or a system either star or delta without any connection to ground, is not a matter of very great importance; and since no theoretical con- ELECTRICAL PRINCIPLES AND CALCULATIONS 121 elusions based on general principles have been arrived at, the engineer is compelled to consider each particular case on its own merits, and be guided by practical results obtained under similar conditions. With a view to eliminating the third harmonic and its multiples, and so obtaining as nearly as possible a sine wave of e.m.f., the generators are usually Y connected, a practice which has the further advantage that the neutral point can be readily grounded if desired. The low-tension windings of the transformers, both at generating and receiving ends of the line are generally delta connected; but so far as the high-tension windings are concerned, these may be star at both ends, or delta at both ends, or star at one end and delta at the other. Then again, the neutral point of a high-tension system may be connected to ground either directly or through a resistance, the results being by no means the same in the two cases. The object of grounding the neutral of a high-tension system is mainly to protect the insulation from abnormally high pres- sures which might aggravate the trouble in the event of a ground occurring on one wire, and so lead to serious interruption of service. It is, in fact, the question of line insulation considered in connection with continuity of service which is generally the determining factor in deciding whether or not the neutral shall be grounded, and whether the grounding, if adopted, shall be with or without the intervention of a resistance. Other considerations, such as the effect on neighboring tele- phone lines, both under normal conditions of working and when a breakdown occurs, will also influence the decision; and it is hardly possible in this place to add much to what has already been said in Article 7 of Chapter II; there is no general rule to be followed seeing that the circumstances of each individual case will have a bearing on the settlement of this question. There would appear to be no particular object in grounding well-insulated systems of moderate pressures up to, say, 60,000 volts; but, in the large high-pressure transmissions, especially with an extended system of branch circuits and tie lines, a dead- grounded neutral is usually desirable. Any three-phase trans- mission system on which the insulation is likely to give trouble when the pressure to ground is raised in the proportion of \/3 : 1 should have the neutral grounded, either directly or through a suitably proportioned resistance. 122 ELECTRIC POWER TRANSMISSION 66. Interference Between Power and Telephone Lines. Al- though the question of interference between power lines and neighboring telephone lines is a very important one, it is also a difficult one to settle satisfactorily or even discuss adequately in a book dealing primarily with the design of high tension trans- mission lines. The problem is of particular interest to the tele- phone engineer who will no doubt ultimately find a satisfactory remedy for the very real troubles which are liable to occur especially when abnormal conditions lead to unbalancing of the power load when telephone wires run parallel to alternating- current power lines for a considerable distance. It is a comparatively easy matter to calculate the flux of induction which the current in the power conductors will set up in the loop formed by the telephone wires, and by carefully planned and frequent transpositions, this effect can be greatly reduced if not entirely overcome; but the electrostatic effects are probably of greater importance because they are less easily dealt with. 67. Insulation of Telephone Lines. It has only lately been realized that one of the essential requirements for telephone lines strung on the same supports as, or very close to, high-pressure power conductors, is high insulation. This good insulation is necessary to prevent puncture of the insulators when high poten- tials are induced on the telephone wires at times of abnormal conditions such as intermittent short circuits, or lightning disturbances on the power lines. As an example of good practice in this respect, the Georgia Railway & Power Co. have provided insulators suitable for a working pressure of 22,000 volts to carry the telephone wires which parallel their 110,000- volt power lines from Atlanta to Tellulah, Ga. 68. Electrostatic Induction. The dielectric field due to the alternating voltage of the power conductors induces a varying charge upon the neighboring telephone wires. If each wire of the telephone line were at an equal average distance from each conductor of the power line, there would be no difference of poten- tial created between the two sides of the telephone receiver, and there should be no buzzing, etc., due to this cause. In other words, with adequate, properly worked out transpositions, the capacity currents passing between the power line and the tele- phone line will not pass through the telephone receiver. But, even if the electrostatic flux is at all times of the same kind and ELECTRICAL PRINCIPLES AND CALCULATIONS 123 amount for both wires of a telephone circuit, this does not prevent the telephone circuit as a whole being subject to alternating pressures relatively to ground, and these pressures may reach high values, depending upon the voltage of the power line and the prox- imity of the two (parallel) circuits. If the telephone circuit is grounded, the charging current passing between the power con- ductors and the telephone line will find its way to ground by flowing along the telephone wires, and since this current may amount to several amperes, trouble is almost certain to occur unless what are known as "drainage coils" are pro- vided. If a choke coil with an iron core such as the primary of an ordi- nary lighting transformer is con- nected across the two wires of the telephone circuit, and then has its middle point connected to ground, the electrostatic charge will be "drained" off the line without in- terfering with the operation of the telephone. The telephone line par- alleling the 110,000- volt transmission of the Georgia Railway and Power Co., 1 is provided with 15-k.w. stand- ard 2200-Volt distribution trans- FIG. 42. Telephone wires on . , . , j \ e same pole as single phase power formers (with open secondaries) for circuit. this purpose. 69. Magnetic Induction. Referring to Fig. 42, the voltage induced by the single-phase power circuit AB in the loop formed by the wires C and D of the parallel telephone circuit may be calculated as follows if we assume that there are no transposi- tions and that the current wave is a pure sine curve. By formula 25 (Art. 43) the flux in the loop CD due to the conductor A carrying a current 7 is and the flux due to the current / in the conductor B is 1 Refer to interesting article by E. P. Peck in Electrical World, Sept. 9, 1916, Vol. 68, p. 515. 124 ELECTRIC POWER TRANSMISSION the total flux being, By making the substitutions and alterations as in obtaining formula (27) we get, Volts induced per mile run of the j = Q Q , /oA\ (53) two parallel circuits \a c oi" Length of string Dry flash-over, K. V. Wet flash-over, K. V. 80^' 49 20H" 160 103 30 H" 239 157 41" 311 211 a$i*' 265 61>" 440 319 26,000:ib. Net weight, Ib 31 62 93 124 155 186 78 g Figs. 53 and 54 are sectional views of suspension insulators manufactured by the R. Thomas and Sons Co. of East Liverpool, Ohio; and Fig. 55 is an insulator made by the Ohio Brass Co. of Mansfield, Ohio, as originally selected for the 140,000-volt transmission lines of the Au Sable Electric Co. (Au Sable to Battle Creek, Mich.), a string of 10 units being used on this voltage. 142 ELECTRIC POWER TRANSMISSION A great deal of attention has recently been devoted to the im- provement of the suspension type of insulator. The parts of the early designs were cemented solidly together, with the result that the differential expansion of porcelain and metal caused cracking and consequent destruction of the procelain after being in use for only a few years. That experts in insulator design realize the effects of expansion and other causes leading to rapid deterio- ration of porcelain insulators is evidenced by the experimental work that has been carried on, and the number of papers pub- lished, relating to this subject, during the last two or three years. c > FIG. 55. Suspension type insulator. Fig. 56, reproduced from a drawing kindly supplied by the Locke Insulator Manufacturing Co., shows how allowance has been made for expansion. An insulator unit as shown in Fig. 56 may be used in both vertical and horizontal positions. When heavy strains have to be resisted, it is often preferable to use two or three strings of insulators in multiple. A good design of suspension unit can be made to have a breaking strength of about 6000 lb., but for greater mechanical strength some of the best features of design from the electrical viewpoint may have to be sacrificed. An interesting discussion of the modern line insulator will be INSULATION OF OVERHEAD LINES 143 found in the Paper by Mr. G. V. Twiss in Vol. 55, No. 267 (June, 1917) of the Journal of the Institution of Electrical Engineers. It is generally true that American engineers are the acknowledged experts in high-tension transmission work, and so far as overhead lines in England are concerned, these are so few and of such low voltage that they need not be considered. But British engineers have carried out important developments in India, Africa, and other Colonies abroad, and such work, although not invariably based on American practice, is always carefully planned and exe- cuted. The differences between British and American methods are worthy of some study, if only to put us on our guard against the tyranny of "established practice." FIG. 56. Suspension type insulator. Line voltage = 16,000. Leakage distance = 12 in. Striking distance = 3 J in. 75. Wall and Roof Outlets. When overhead high-tension con- ductors have to be brought into buildings, the design of insulating bushings should receive careful consideration. The fact that every bushing acts as a condenser or series of condensers between wire and ground, must steadily be recognized just as in the case of the pole-line insulators, except that the effect is even more marked in connection with bushings entirely surrounding the conductor than with insulators that support the wire at one end only. When the climate and weather conditions are favorable, it is well to avoid bushings entirely. In such cases, the wires cannot be brought down through the roof of the building, but 144 ELECTRIC POWER TRANSMISSION they must enter at the side; a suitable protecting hood or roof being placed above the wires on the outside of the building. The smallest dimension of the opening in brick, stone, or concrete wall should preferably not be less than as given below: For line pressure, volts 22,000 33 ; 000 44,000 66,000 88,000 110,000 On each side of the wall opening, the conductor is carried by line insulators, of the pin or suspension type, as the voltage may Fio. 57. Porcelain entering bushing for 70,000 volts. require, these being so arranged as to maintain the conductor in the center of the opening, with a slight downward incline toward the outside of the building to prevent rain-drops being carried to the inside. When bushings are used, these must necessarily be thicker at or near the center, where the ground potential is brought up close to the conductor, than at either end. Fig. 57 shows a porcelain bushing built up of four parts, as supplied by the R. Thomas & INSULATION OF OVERHEAD LINES 145 Sons Co., suitable for a working pressure of 70,000 volts. This bushing weighs about 100 Ib. and will flash over with 170,000 volts (dry) and 138,000 volts (wet). Fig. 58 shows a floor bushing suitable for a working pressure of 44,000 volts. Its dry flash-over voltage is 112,000, and wet flash-over 75,000. One method of bringing the high tension conductor through the wall of a building is illustrated by Fig. 59, which shows the overhead wire anchored to the wall by a string of suspension insulators, and passing through a standard porcelain wall bush- ing as manufactured by the Ohio Brass Co. Fia. 58. Floor bushing. In the case of a roof bushing, permitting of the conductor passing vertically downward to the inside of the building, a hollow elongated barrel-shaped insulator (that is, of greater diameter at center, where it is supported on the outside, than at the two ends) filled with an insulating substance of higher specific induct- ive capacity and greater dielectric strength than air, makes a satisfactory arrangement provided there is no danger of the oil or other insulating filling leaking out of the containing shell. A high- grade insulating oil will frequently give good results, but it is liable to leak out at the joints, and, moreover, the use of oil calls for a 146 ELECTRIC POWER TRANSMISSION larger and heavier bushing than if the filling has a specific ca- pacity more nearly equal to that of the porcelain shell (porcelain being the material most generally used). It will be understood Hood should be used where. Icicles can fall on Insulators., from Eaves FIG. 59. that if the substance filling the space between metal conductor and hollow bushing has the same specific capacity as the material of the bushing, there will be no change in the potential gradient at the inner surface of the bushing. The thickness of the porcelain shell and the distance between inner surface of shell and conductor surface should be proportioned in ac- cordance with the insulating material intended for use as a filling. 76. Design of Insulating Bushings. Without attempt- ing to go into the details of de- sign, the application of the principles and formulas of Ar- ticle 72 may be illustrated by considering the stresses in the insulation which separates a cylindrical rod at high potential from a concentric cylindrical tube at ground potential. Fig. 60 is a section through a con- ductor of radius r separated by insulating material of specific Fio. 60. Section through insulating bushing. INSULATION OF OVERHEAD LINES 147 inductive capacity k from a concentric metal cylinder of radius R. The equipotential surfaces will be cylinders, and the flux density over the surface of any cylinder of radius x and of unit \fr length, say 1 cm., will be D = ~ -- By formula (57) the potential gradient is, In order to express this relation in terms of the total voltage E, it is necessary to substitute for the symbol ^ its equivalent E X C, and calculate the capacity C of the condenser formed by the rod and the concentric tube. Considering a number of con- centric shells in series, the elastance 1 may be written as follows: i- r ^ C J 2irxKk dx 1 , R /rr . x e - (59) Substituting in (58), we have E volts per centimeter . . the maximum value of which is at the surface of the inner con- ductor, where -- (61) This formula is of some value in determining the thickness of insulation necessary to avoid overstressing the dielectric; but it is not strictly applicable to wall bushings in which the outer metal surface is short as compared with the diameter of the opening. The advantage of having a fairly large value for r is indicated by formula (61), and a good arrangement is to use a hollow tube for the high-tension terminal. Solid porcelain bushings with either smooth or corrugated surfaces may be used for any pressure up to about 40,000 volts. In designing plain porcelain bushings it is important to see that the potential gradient in the air space between the metal rod and 1 The reciprocal of the permittance or capacity. 148 ELECTRIC POWER TRANSMISSION the insulator is not liable to cause brush discharge, as this would lead to chemical action, and a green deposit of copper nitrate upon the rod. Fig. 61 shows a bushing that is liable to give trouble owing to brush discharge in the air space between the high-tension rod and the inner surface of the porcelain bushing. The reason for this will be best understood by working out a numerical example. Let us assume the following numerical values: Radius of inner conductor, r = 0.25 in. Inside radius of porcelain bushing, r' = 0.375 in. Outside radius of porcelain bushing (in contact with grounded metal cylinder) R = 1.5 in. Dielectric constants: for air k = 1; for porcelain k = 4.5. FIG. 61. Type of bushing liable to cause brush discharge on surface of wire. We have here the case of two capacities in series, the first between the high-tension rod and the inner surface of the porcelain, with air as the dielectric, and the second between the inner and outer cylindrical surfaces of the porcelain. The dielectric flux (and the charging current) being the same in both, and since ^ = E X C, it follows that the potential difference across each con- denser will be inversely proportional to the capacity, or directly proportional to the elastance as given by formula (59). Let E a be the voltage across the air gap, and E p the voltage across the porcelain sleeve : then, w log () X 4.5 log = 4.5 E p , /R\ . /1500\ log (F) log (375) which shows the voltage across the small air space to be greater than that across the greater thickness of the porcelain bushing. INSULATION OF OVERHEAD LINES 149 The voltage gradient which will cause corona or brush discharge on the surface of a rod of radius r, according to Mr. Peek, 1 is G v 31 ( 1 H '~/= ) where r is in centimeters. In this example, with r = 0.25 in. we find the value of G v to be 42.7 k.v. per centi- meter. Then, by formula (61) the maximum permissible value of the voltage across the air space is E a = 42.7 X 0.25 X 2.54 X log e (^~) = H k.v. The total permissible pressure across the bushing is therefore 11 X (l + i~3i5~) = 19 - 37 k - v - of which the virtual value is 19.36 /=-= 13.67 k.v. Let us now consider the stress at the surface of the conductor if the porcelain sleeve is entirely removed, leaving air as the only dielectric between the rod and the outer metal cylinder. By formula (61), max. value of E = 42:7 X 0.25 X 2.54 X log e (Q^) = 48.6 k.v. of which the virtual value is 48.6/\/2 = 34.4 k.v. Thus, if the porcelain bushing is entirely removed, and re- placed by air, the voltage (in this example) may be increased from 13.67 to 34.4 k.v. before corona will form at the surface of the rod. Two remedies for the trouble resulting from the original design suggest themselves. We might (1) coat the inside of the porcelain bushing with tin-foil and connect this electrically with the high-tension rod, or (2) we might fill the intervening air space with oil or some solid insulating compound of higher spe- cific inductive capacity than air. (1) This method is equivalent to making the high-tension con- ductor of radius 0.375. Let us assume the total voltage to be 34.4 k.v. as calculated for the arrangement with air space only. The maximum potential gradient, by formula (61), will then be, G max . = - 34.4 X 1.41 ^^ = 36 g k y per centimeter 0.375 X 2.54 X log c (Q^) 1 "Dielectric Phenomena," by F. W. Peek, Jr., McGraw-Hill Book Co. 150 ELECTRIC POWER TRANSMISSION which is less than with the smaller diameter of rod, notwithstand- ing the reduced thickness of the dielectric. (2) If we assume, for simplicity in the calculation, that the filling compound has the same dielectric constant as the porcelain (k = 4.5), we may write, 34 4 X 1 41 (j m ax. = - ' - '- - =-= = 42.7 k.v. per centimeter 0.25 X 2.54 X log which is, of course, the same as when the entire space is filled with air; but there will be no corona formation on the surface of the high-tension conductor. The disadvantage of small diameters for high-tension conductors is well brought out by this example. Without any increase in the size of the outer (grounded) cylinder, an increase in diameter ofr the conductor from % in. to % in- has reduced the voltage gradient at the conductor surface (where its value is a maxi- mum) from 42.7 k.v. to 36.8 k.v. per centimeter, and this not- withstanding the fact that the distance from metal to metal, is actually less in the case of the lower figure. There is obviously a limit which is easily calculated beyond which any increase in the diameter of the inner rod would lead to break-down on a lower voltage. 77. Condenser Type of Bushing. By separating thin concentric layers of insulating material by tubes of tin-foil or other metal, and so proportioning the lengths of the tin-foil tubes that the areas remain the same, notwithstanding the variations in diameter, it is possible to design a bushing which virtually consists of a number of condensers of equal capacity all connected in series. In this manner the potential gradient can be made uniform throughout the thickness of the insulating bushing. Commercial entering bushings and transformer terminals have for some time past been built on this principle. The design of such terminals is not quite so simple a matter as this brief reference to the main principle involved might suggest, but the commercial insulators of this class have, on the whole, given good service. The tendency to equalize the potential gradient throughout the concentric layers of insulating material, leads to a bushing of smaller overall diameter than when no attempt is made to equal- ize the stresses. Thus, if the figure of 36.8 k.v. per centimeter as calculated in the numerical example of the preceding article INSULATION OF OVERHEAD LINES 151 is assumd to be the maximum working stress of a solid porcelain bushing of inside radius % in. and outside radius 1^ in., a bush- ing of the condenser type designed for the same maximum stress and suitable for the same total voltage, would have an outside radius of only 3 34.4 X 1.41 R = 8 + 36.8 X 2.54 = ' 893 m This calculation does not take account of the thickness of the dividing layers of tin-foil, or of the fact that perfect equalization of the stress is not obtainable in a practical design of bushing. Before considering such matters as factors of safety, spacing between wires, and probable limits of pressure on power transmis- sion lines, it will be well to review briefly what is known about brush' discharges and corona formation, which are liable to become important factors when transmitting energy at the higher voltages. 78. Formation of Corona, and Accompanying Losses of Power. When the pressure on an overhead transmision system exceeds a certain critical value depending upon the spacing and diameter of the wires, there will appear on the surface of the con- ductors a halo-like glow to which the name "corona" has been given. Apart from this luminous effect, the appearance of the corona is accompanied by a certain loss of power proportional to the frequency and the square of the amount by which the pres- sure between conductors exceeds a certain value known as the disruptive critical voltage. If the distance between outgoing and return conductors is comparatively small (less than fifteen times the diameter of the wire) there will be a spark-over when the dis- ruptive critical voltage is reached; but with the greater separation such as occurs on practical high-tension transmission lines, the effect of the high potential at the conductor surface is to break down the resistance of the air in the immediate neighborhood of the conductor surface. A luminous cylindrical coating of air, acting as a conductor of electricity, is thus formed, the di- ameter of which will depend on the amount by which the actual value of the applied potential difference between wires exceeds the disruptive critical value of the potential difference. The result is equivalent to an increase of the diameter of the conduct- ors, thus raising the value of the voltage necessary to break down new concentric layers of surrounding air, until it is approximately 152 ELECTRIC POWER TRANSMISSION equal to the voltage impressed on the wires. During the last few years much light has been thrown on the formation and effects of the corona. Among the earlier workers in this field were C. F. Scott, Harris J. Ryan, J. J. Thomson, H. B. Smith, Signer Jona, Lamar Lindon, E. A. Watson, and, among the later investi- gators, J. B. Whitehead 1 and F. W. Peek, Jr. 2 Suppose a cylindrical wire of radius r is surrounded by a con- centric metal cylinder of internal radius R and that visible corona starts when the observed voltage is E v , then the maximum poten- tial gradient at the surface of the wire, by the previously developed formula (61) is (62) (D The formula for parallel wires is Ev (63) * $ where d is the distance between centers of wires. The value of G v (the apparent strength of air) is found to be independent of the spacing between wires, but it is not found to be independent of their diameter. The apparent strength of air is greater at the surface of small than of large wires. Mr. Peek has found that, at a distance from the surface of any cylindrical wire equal to 0.301 \/r cm., the breakdown gradient of all sizes of wire is the same, namely about 30 k.v. per centimeter. This leads to the formulas which will be found at the end of this article. The losses which occur through corona are not different in kind from I 2 R losses; but since they occur whenever a visual corona appears, and may reach high values if the line voltage appreciably exceeds the voltage at which corona is first formed, the importance of designing high tension lines so as to avoid excessive corona formation is evident. The more important features and effects of the corona of in- terest to the practical engineer may be summarized as follows: 1. The loss due to leakage of current from the conductor into 1 Proc. A. I. E. E., vol. xxix, p. 1059 (1910), and later contributions. z Proc. A. I. E. E., vol. xxxi, p. 1085 (June, 1912). Also Journal Franklin Institute, Dec., 1913; and Book "Dielectric Phenomena," McGraw-Hill Book Co., 1915. INSULATION OF OVERHEAD LINES 153 the surrounding air is practically negligible for pressure values below the disruptive critical voltage; no account need be taken of such leakage on alternating-current circuits operated at pres- sures below 44,000 volts, unless the wires are at high altitudes. On 80,000 volts, however, the loss may be appreciable, and a visible corona may even be formed if the wires are small in diameter. 2. The current passing from the wires into the air on an alter- nating system is an energy current in phase with the pressure. 3. On alternating-current systems, the critical break-down voltage will depend upon the maximum value of the e.m.f. wave, and therefore on the "form factor." 4. The break-down voltage or, more properly, the disruptive critical voltage is determined by the potential gradient at the conductor surface; it is therefore dependent upon the diameter and spacing of the wires; being higher with the larger diameters and spacings; it is also dependent upon the density of the air, and therefore on the temperature and barometric pressure. 5. The loss of power due to corona formation is approximately proportional to the frequency (within the usual commercial range), and to the square of the excess of line voltage over the disruptive critical voltage. 6. The disruptive critical voltage is the voltage at which the disruptive voltage gradient of the air is reached at the conductor surface. It is highest when the conductor surface is smooth and quite clean. It is lowered by roughness or dirt on the conductors, also by smoke and fog in the atmosphere, sleet on wires and falling sleet, rain and snow storms, and low barometric pressure. All these causes tend, therefore, to increase the corona losses. 7. The visual corona occurs only at a pressure above the dis- ruptive critical voltage and is an indication that there is loss of power in the air. When considering the effects of the corona formation on over- head wires, it is convenient, as in the case of the majority of electrical problems connected with transmission lines, to consider each wire separately in relation to the neutral plane or line. Since the formation of the corona depends upon the electric stress at the surface of the conductor, it is the potential gradient in the immediate neighborhood of the wire which, as previously men- tioned, is the determining factor in corona formation. The dis- 154 ELECTRIC POWER TRANSMISSION ruptive critical voltage for any particular wire under specified atmospheric conditions will, as previously mentioned, depend upon the diameter of the wire and the distance of the wire or wires forming the return conductor; also upon the density of the air and, to some extent, upon the surface condition of the wire. Mr. Peek's formula is: E = 21.lm r8 log, - k.v. to neutral (virtual value) (64) in which r = radius of conductor in centimeters, d = distance between centers of the outgoing and return (par- allel) conductors, in centimeters, m = a factor depending upon the surface condition of the con- ductor, = 1 for polished wires, = 0.98 to 0.93 for roughened or weathered wires, = 0.87 to 0.83 for stranded cables (average = 0.85), 5 = a factor depending on the air density, 3.926 273 + t in which b is the barometric pressure in centimeters of mercury, and t is the temperature in degrees Centigrade. The luminosity, or visible halo of light surrounding the con- ductor, does not occur until a higher pressure has been reached, the increase over the critical disruptive voltage being dependent upon the diameter of the conductor. Mr. Peek's formula for the visual critical voltage (kilovolts to neutral) is: E v = 21.lm.r5 l + r5 (l + ^=\ log, j[ (65) where the surface factor m v has the same value as m for wires, and may be taken at 0.82 for a decided visible corona on seven- strand cables. The notation is otherwise as above. The formula for loss of power in fair weather, in kilowatts per kilometer of single wire, as given by Mr. Peek, is: P = ^ x (f + 25) X-Jj X (E n - E Y X 1(T 5 (66) where / is the frequency in cycles per second, and E n is the actual (r.m.s.) pressure between wire and neutral, expressed in kilo- volts. The approximate loss under storm conditions is obtained INSULATION OF OVERHEAD LINES 155 by taking E as 80 per cent, of its (virtual) value as calculated by formula (64). The transmission line engineer will usually prefer formulas with inch units and common logarithms; and the following may be used: Eo = 123 m r8 Iog 10 k.v. to neutral (67) E v = I23m v r5 (l + ^4fyogw/^ k.v. to neutral (68) r p = - (/ + 25) (E n - E Y 10- 5 kw. per mile of single conductor (69) The air density factor can, if desired, be calculated by the formula = 17.96 459 + t where b = barometric pressure in inches of mercury, and t = temperature in degrees Fahrenheit. (Note that when 6 = 29.9 and t = 77 degrees, 5=1.) As a guide in estimating the average pressure at high altitudes, the following figures may be used : Elevation, sea level, b = 29.9 2,000 ft., 6 = 27.6 4,000 ft., b = 25.6 6,000 ft., b = 23.7 8,000 ft., & = 22.0 10,000 ft., 6 = 20.4 12,000 ft., 6 = 18.9 As a practical example of corona losses, consider a 100 mile, three-phase, 110,000 volt, 60-cycle transmission, using No. 1 seven-strand conductors spaced 6 ft. apart. Calculate the approximate fine-weather corona loss if the air density factor is unity (5 = 1). By formula (67) Eo = 123 X 0.85 X 0.165 X log(V^) = 45.6 k.v. XU.lOO/ By formula (69) = 5.1 kw. per mile of single conductor. 156 ELECTRIC POWER TRANSMISSION The total loss to be expected under fair weather conditions is therefore 5.1 X 3 X 100 = 1530 kw. The assumption is here made that the line voltage (and there- fore E n ) remains constant over the entire length of 100 miles. 79. Corona Considered as " Safety Valve" for Relief of High- frequency Surges or Over -voltage Due to Any Cause. The loss, as calculated in the above example, is not small; and since it is proportional to the square of the excess of pressure over the disruptive critical voltage, a small increase of pressure will lead to an enormously increased dissipation of energy in the air. Thus, if the pressure of 110 kv. in the above example be supposed to increase only 10 per cent, the total dissipation of power, instead of being 1530 kw., would be 2800 kw. It has indeed been stated that on the 110,000-volt system of the Grand Rapids- Muskegon. Power Co., the line loss due to corona discharge actually increases 100 per cent, for a 10 per cent, rise in pressure. This property of the corona suggests the possibility of working high-voltage transmission lines at a normal pressure in the neigh- borhood of the critical disruptive voltage where the loss would be inappreciable. An extra-high-voltage discharge, due either to atmospheric lightning, or to internal causes, would then be largely dissipated in the corona itself. This may, to some extent, account for the fact that fewer lightning troubles are experienced on the very high voltage transmissions than on the lower voltage lines. The insulation of the conductors being such as to with- stand, without breakdown, pressures considerably in excess of the disruptive critical voltage of the corona, a large amount of oscillating energy can be dissipated in the air before the voltage rises to such a value as to pierce or shatter insulators or damage apparatus connected to the line. On the other hand, too much reliance should not be placed on the corona as a means of dissi- pating large amounts of suddenly impressed energy; because lightning and similar disturbances, being to a great extent local, must discharge their power locally, and the corona losses over a short section of the transmission line cannot under any circum- stances be very great. 80. Spacing of Overhead Conductors. It is difficult to lay down rules for the proper spacing of overhead conductors. The question has been settled in the past by the individual engineer who has usually striven to be "on the safe side" in the matter of possible discharges between wires under abnormal conditions INSULATION OF OVERHEAD LINES 157 such as strong and variable winds. The result is that great differences are to be found in the wire spacings in different coun- tries or on different transmission systems in the same country. The spacing of the conductors should be determined by consid- erations partly electrical and partly mechanical. With the longer spans, the spacing should be greater than with short spans, 10 20 30 40 50 60 70 80 90 100 110 120 130 140 Line Pressure (between Wires.) -.Kilov.olts FIG. 62. Approximate separation of overhead conductors. apart from voltage considerations. The material and diameter of the conductors should also betaken into account when deciding upon the spacing, because a small wire especially if of aluminum having a small weight relatively to the area presented to a cross wind, will swing out of the vertical plane farther than a conductor of large cross-section. Usually wires will swing synchronously in a wind; but with long spans and small wires, 158 ELECTRIC POWER TRANSMISSION there is always the possibility of the wires swinging non-syn- chronously, and the size of wire, together with the maximum sag at center of span, are factors which should be taken into account in determining the distance apart at which they shall be strung. A horizontal separation equal to something between one and one and a half times the sag at the temperature corresponding to the season of highest wind velocities should be sufficient to prevent wires swinging within sparking distance of each other; the closer spacing being used with copper conductors of large diameter. The curves of Fig. 62 will be found to give spacings generally in accordance with present-day practice. These figures may be used as a guide in arriving at a suitable value for the horizontal spacings. The vertical spacing may be less; but it is usually undesirable to suspend wires in the same vertical plane, especially in locations where sleet and ice deposits are likely to occur. Distance Between Conductor and Pole or Tower. The following clearances are recommended: Line pressure, k.v. 10 (and under) 9 15 10 22 11 35 14 44 17 66 00 24 oo OO 110 Kfi 36 In the case of suspension type insulators it is well to arrange for the clearance, even under conditions of greatest deflection caused by high winds, to be not less than the sparking distance over the string of insulator units. 81. Practical Limitations of Overhead Transmission-line Voltages. From the foregoing review of the insulation prob- lems to be met with on long-distance overhead transmissions, it will be clear that manufacturers are now in a position to pro- vide insulation amply sufficient for present requirements. Power is actually being transmitted at 150,000 volts. The lines of the Au Sable Electric Co. in Michigan, transmitting power at INSULATION OF OVERHEAD LINES 159 140,000 volts, consist of stranded copper conductors about %-in. diameter on 500-ft. spans, with a sag allowance of approximately 12 ft. The shortest distance between conductors is 12 ft., this being the vertical height between the two conductors on one side of the steel supporting towers. There is practically no visible corona, but a buzz or hum, due no doubt to brush discharge, can be heard in the neighborhood of the transmission lines. Although there are no insurmountable difficulties in providing ample insulation for these high voltages, it is the engineer's business to provide such insulation as will be justified by economic considerations. It is also his business to determine the voltage of transmission on the same basis, and resist the temptation to experiment in voltages higher than may be justified by commer- cial considerations. It is well to bear in mind that the economical transmission voltage depends not only on the length of the line, but also on the amount of power to be transmitted; and although a 200,000- volt transmission offers no serious engineering difficulties, the conditions under which a transmission at so high a voltage would be a commercial success, are very seldom found. 82. Factors of Safety : Rating and Testing of Line Insulators. When selecting insulators and deciding upon the spacing and arrangement of conductors suitable for a given voltage, the factor of safety to cover abnormal pressure-rises is a matter of great importance, since it is obviously bad engineering to provide insulation in excess of what experience has shown to be a reason- able safeguard against interruption of service. Generally speaking, the insulators should, when dry, withstand a pressure test of 2^2 to 3 times the working pressure to ground, applied for five minutes, and a wet test of not less than twice the working pressure. This would sometimes be considered too small a margin of safety; but the ratio between test pressure and working pressure will depend upon whether the line voltage is high or low. The following safety factors, representing ratio between wet-test pressure and working pressure are generally in accordance with usual practice; but the engineer should use his judgment in a matter of this sort. It is clear that, on the coast, where gales and salt sea mists are prevalent, the factor of safety should be rather higher than in a district where the climatic conditions are more favorable. The effects of high altitude to be referred to later must also be taken into account, 160 ELECTRIC POWER TRANSMISSION Working pressure (voltage between line wires) Safety factor (wet test) 20000 40000 2.5 2.2 80000.... 2.0 Above 80000 volts 1 8 to 2 As the wet or "rain" test will give different results, depending on the method of conducting the tests, there should be a clear understanding between the purchaser and manufacturer on this point. A very common specification is that the spray shall be directed at an angle of 45 degrees, under a pressure of 40 Ib. per square inch at the nozzles; the flow being regulated to give a precipitation of 1 in. in 5 minutes. The method of attaching test wire and ground connection to the insulator should also be clearly defined. The test pressure is usually measured by means of a spark gap; and the alternating e.m.f. used should conform as nearly as possible to the sine wave . The striking distances in air between No. 3 sharp needles, as given by the Locke Insulator Mfg. Co., are as follows: Kilovolts Kilovolts 20 | 1.00 200 20.50 40 2.45 225 23.05 60 4.65 250 25.60 80 7.10 275 28.30 100 9.60 300 31.00 125 12.25 350 36.10 150 15.00 400 41.20 175 17.80 It will be observed that for pressures above 100,000 volts the gap between needle points is approximately 1 in. per 10,000 volts. The pressures referred to in the table are the virtual or root- mean-square values of the test pressure, on the sine wave assumption. Effect of Altitude. The insulators on a transmission line erected at high altitudes will flash over with a lower voltage than if the line were erected at sea level. The reason for this is the reduced air pressure at the higher elevation. The flash-over voltage will not be exactly proportional to the barometric pres- INSULATION OF OVERHEAD LINES 161 sure because the electrostatic field is not uniform, but depends upon the type and design of the insulator. Mr. Peek gives some results of experimental work on various standard designs of insulator in his book on Dielectric Phenomena previously referred to; but since the departures from the theoretical relation for uniform dielectric fields is very small, the correction for altitude can safely be made by assuming the flash-over voltage to be directly proportional to the barometric pressure. As an example: suppose the flash-over voltage of a pin type insulator is found to be 100 kv. on test at sea level; then, if used on a transmission line erected at an elevation of 8000 feet, it would be liable to flash over with a pressure of only 100 X 22 299 = 73 '^ k y * (The proportional figures for air density are taken from the table on page 155.) It is thus easy to de- cide upon a factor of safety which shall make proper allowance for the elevation at which the insulator may have to operate. Rating of Insulators. The fact that conditions of tests and factors of safety may have to be modified in accordance with the conditions under which the insulator will be expected to operate, suggests the subject of insulator rating. At the present time this question is far from being settled in a satisfactory manner, and it would seem that manufacturers and users should get together with a view to deciding upon an acceptable basis which would permit of any design of insulator being placed in a particular class, and so facilitate the comparison between tenders submitted by competing firms for insulators to operate under given conditions. At first sight the most reasonable method appears to be to adopt some arbitrary rule depending upon the dry and wet flash-over voltage of the insulator: for instance, to specify that for the higher line pressures the dry flash-over should be three times the line voltage, and the wet flash-over twice the line voltage. This would, however, probably lead to confusion because of the number of variable factors involved in determining the dry and wet flash-over. It is possible as suggested to the writer by an engineer connected with one of the best known firms manufacturing porcelain insulators that a method of rating depending not upon tests, but upon design dimensions, such as the overall height and the minimum arcing distances (both wet and dry), might prove a useful basis of classification. 162 ELECTRIC POWER TRANSMISSION Detecting Faulty Insulators while Line is in Operation. Another question which is now receiving attention, but is not settled or standardized, is the best means of detecting faults in insulators while in use. It is only of recent years that accurate data on the "life" of high tension insulators is becoming available; and the continued action of alternate heat and cold, dryness and dampness, on the porcelain or rather on the complete assembly of porcelain, metal, and cement is found to necessitate a very large percentage of replacements after a line has been in operation many years. The causes of rapid deterioration especially after several years of service are being investigated, and eliminated as far as possible in the later designs; 1 but in the meanwhile, lines that have been in use for a considerable time are giving trouble in the matter of insulation, involving increasing vigilance and activity on the part of the operating staff. There is room for improvement in the methods now available for detecting in- cipient faults in line insulators without interrupting the supply or disconnecting insulators from the live wires. One method, which involves the use of a telephone, and is said to give reliable information as to the condition of the insulators, is described on page 821 of Vol. 64 of the Electrical World (Oct. 24, 1914). The reader who is interested in this matter of insulation troubles should also refer to an article by Prof. H. J. Ryan in the Journal of Electricity (San Francisco), Feb. 27, 1915; and to the article "Testing Insulators to Assure Continuous Service" by Professor R. W. Sorensen in the Electrical World of Sept. 1, 1917 (Vol. 70, p. 426). 1 Refer Trans. A. 1. E. E., vol. xxxiii (1914), pp. Ill and 119 (J. A. Brun- dige), and p. 1731 (A. O. Austin). Vol. xxxiv (1915), p. 465 (E. E. F. Creighton). Vol. xxxvi (1917), p. 527 (W. D. Peaslee), p. 535 (J. A. Brundige), and p. 545 (A. O. Austin). CHAPTER VI PROTECTION AGAINST LIGHTNING TRANSIENT PHENOMENA 83. Theoretical Considerations. Before taking up the matter of lightning disturbances and the means adopted for minimizing their destructive effects on line insulation and station apparatus, the causes leading to surges, oscillations, and travelling or stand- ing waves, will be very briefly discussed. No attempt will be made to deal thoroughly with this somewhat difficult subject which has lately come into prominence because of the increasing distances to which energy is being transmitted electrically. Nothing will be included here beyond the elementary considera- tions with which the engineer engaged on the design of high- tension long-distance transmission lines should be familiar. For a complete study of the principles underlying transient phenomena energy surges and oscillations, and the peculiarities of hypothetical "quarter wave length" transmission lines the reader is referred to authorities such as Dr. Steinmetz 1 and the writings, by various authors, that have appeared recently in the technical Journals and the publications of the Scientific and Engineering Societies. The relation existing between the voltage #nd the current of any transient electrical disturbance occurring in a circuit depends upon the relation between the magnetic flux-linkages per unit current or the inductance and the permittance or electro- static capacity. Consider a circuit in which there is alternating or oscillating energy which is not utilized by any form of receiving apparatus, and is not dissipated in the form of heat through the ohmic resistance of the conductors, or through "dielectric hysteresis" or corona: it is obvious that, at the instant when the current wave passes through zero value, the whole of the energy must be stored in the electrostatic field, and similarly, at the instant when the pressure wave passes through zero value, the whole of the oscillating energy must be stored in the electromagnetic field. 1 "Transient Electric Phenomena and Oscillations." 163 164 ELECTRIC POWER TRANSMISSION Moreover, so long as the interchange of energy from one form to the other continues without diminution of amount, these two quantities must be exactly equal. This conception of the oscil- lations of energy in a circuit having negligible resistance, but appreciable inductance and capacity, is fundamental, and we shall examine it in further detail with a view to arriving at a definite relation between the amplitudes of the voltage and current waves. Energy Stored in Magnetic Field. Since the engineer usually prefers to think of volts and amperes, the product of which repre- sents power (watts) or the rate at which work is being done, we may say that the energy stored in a magnetic field during a short interval of time dt seconds is ei X dt watt-seconds or joules. In this connection the voltage e is the e.m.f . developed in a conductor carrying i amperes when a change in current di causes a change of flux d$ in the short interval of time dt. Thus, since we are considering the flux linking with a circuit of one turn (a trans- mission line conductor) in a medium (air) of constant permeability, we may write, d< T di e== ~dt =L di whence Energy stored in magnetic field during the | _ T di ' . , interval of time dt } " L ~dt X = Li X di and since the current grows from zero to its maximum value in a quarter of a period, we have, Energy stored in magnetic field 1 _ j 1 ^5 Z ^- during one quarter period J = Yz LJ 2 max. joules (70) It is easy to show in a similar manner that the energy stored in the dielectric circuit in one quarter period while the value of e grows from zero to jK max . is % CE z m& ^ where C is the electro- static capacity of the circuit, or portion of circuit considered. Thus, in the case of a pure, undamped, oscillation, when no energy is supplied from the outside to the circuit, or by the cir- cuit to the outside, it follows that L7 2 = CE Z PROTECTION AGAINST LIGHTNING 165 (7D where L is expressed in Henrys, and C in farads. The quantity -/^ is thus seen to be of the nature of a resistance or impedance, and it may be expressed in ohms. It is generally called the natural impedance of the circuit; but the expressions wave impedance (or resistance) and surge impedance (or resistance) are also used to denote the ratio of the oscillating energy. amperes In the case of an overhead transmission line, the approximate value, per mile, of the inductance betwen one conductor and neutral, as given in Article 48, Chapter IV (page 88) is External Inductance, L = 0.000741 log - and the approximate formula for capacity, as given in Article 10 of Chapter II, is farads whence the surge impedance of an overhead transmission line is approximately, g = 138 log - ohms (72) In practical overhead work, the limiting values for the ratio - will probably be 800 and 50; which, when inserted in formula (71), show that the "natural impedance" of an overhead trans- mission line must lie between 400 and 230, or, to be well on the safe side, between (say) 500 and 200 ohms. A knowledge of this quantity renders it possible to determine the maximum value of any surge pressures that can possibly occur on the line due to the sudden interruption of the current. Thus, if the " natural impedance" is 300 ohms, and the instanta- neous value of the current at the crest of the wave is 200, the surge pressure, however suddenly the current is interrupted, cannot possibly exceed 200 X 300 = 60,000 volts; because this is the maxi- mum value of the pressure wave necessary to store in the elec- tric field the whole of the energy stored in the magnetic field at 166 ELECTRIC POWER TRANSMISSION the moment when the current was interrupted. It is safe to say that, on a practical transmission line, the surge pressure is never likely to exceed 200 times the current in amperes; but, with heavy currents, this may well be sufficient to break down insulation and cause considerable damage to power plant. It must not be over- looked that it is often more difficult to handle heavy currents at comparatively low pressures than small currents at the very highest pressures yet attempted. When the current is large, the opening of switch or fuse on full-load, or an accident causing a break in the circuit, with or without the formation of an arc across the gap, may lead to insulation troubles on many widely separated parts of the system; but on a high-pressure system, even if the current were as large, the insulation is frequently so good that it will withstand without injury the stress imposed on it by the highest possible value of the surge pressure. In underground cables, the capacity is much larger relatively to the inductance than in overhead systems, and the surge im- pedance, ATV has then a smaller value, which may be about one-tenth of the value for overhead lines; but the transformers connected to transmission systems will always have a surge impedance very considerably higher than that of the line itself. The effect of the ohmic resistance in series with the 'inductive and condensive reactances of a circuit, is to damp out the oscilla- tions by dissipating the energy in the form of I 2 R losses. With a sufficiently high value of resistance in the circuit, surges or oscillations of energy cannot take place. For the condition of massed resistance, capacity, and induc- tance, the critical resistance is R = and if R has a higher value than this, oscillations of energy cannot occur. The case of a transmission line with distributed capacity and inductance is much more complicated, and the mathematical analysis is very difficult; but if the receiving end of a long trans- mission line of negligible resistance is closed through a resistance (line to neutral) of value R = \fe (73) PROTECTION AGAINST LIGHTNING 167 there will be no oscillations of energy resulting from a sudden change of potential. In other words, a surge travelling along the line will be completely absorbed, and there will be no "re- flected" waves. (The proof of this statement will be given in Article 86). If R has a value smaller than that given by formula (73) there will be " reflection" of some part of the impressed energy, and oscillations will occur; but these will gradually decrease in ampli- tude according to the logarithmic law. 84. Frequency of Oscillations. The rate at which the oscil- lating energy will pass back and forth between the magnetic and the dielectric fields is entirely independent of the frequency of the power current in a transmission line. By formula (71) we have E = I^~ (74) but since / may be considered as the charging current of a conden- ser of capacity C with a voltage E across the terminals, we can also write E= 2^fC (75) which is obtained from formula (36) of Article 54, Chapter IV. The value of / obtained by equating (74) and (75) is / = j=. periods per second (76) 27r\/LC which is the periodicity of the oscillations when the inductance and capacity are supposed to be concentrated at a given point. When capacity and inductance are distributed as on -a long-dis- tance power-transmission line, the formula for the frequency of oscillation as developed by Dr. Steimnetz is (77) which is the frequency of resonance and is called the natural frequency of the line. It should be noted that L and C, as in the previous formulas, are expressed in henrys and farads; but since we are dealing with a product, not a ratio, of the two quan- tities, it is the inductance per mile X length in miles, and the capacity per mile X length in miles, that these symbols now stand for. 168 ELECTRIC POWER TRANSMISSION By formula (37) of Article 54, Chapter IV, the approximate value of the product CL for an overhead transmission one mile long is seen to be 1 CL 34700 X 10 6 which, when substituted in formula (77), gives for the natural frequency of an overhead line, (78) where L is the distance of transmission, or length of a single conductor, in miles. 85. Wave Length. The rate of travel of an electric impulse along an overhead wire is approximately the same as the velocity of light, or (say) 186,000 miles per second. Thus, if we imagine < 775 miles- ->i 3100 miles Fio. 63. Diagram showing instantaneous value of current at different points on a long transmission line. an alternating e.m.f . of frequency / = 60 applied to the ends of 186000 a circuit of length = 3100 miles, the maximum value oU of the current wave would occur at the receiving end of the line simultaneously with the maximum value at the sending end, but it would be the crest of a current wave which had left the sending end exactly 3-1$ o second earlier. Fig. 63 is an attempt to indi- cate the travel of the electrical impulse over a wire of great length. The ordinates of the curve show the value of the current at any point along the line at the instant when the impressed sine wave of current has attained its maximum value at the sending end of the line. The wave length, or distance covered by one PROTECTION AGAINST LIGHTNING 169 1 86000 complete wave, is - j - miles, which in this example is 3100 miles. A quarter wave length line with a frequency of 60 would be 775 miles long; it would have the peculiarity that the pressure (or current) wave would have zero value at one end of the line at the same instant of time as its value was a maximum at the other end. The characteristics of such a line are quite different from those of an ordinary transmission line, and al- though at ordinary frequencies trouble from this cause is not likely to result, it is possible to get the resonance effect of a quarter wave line with the higher harmonics of a distorted wave, even on practical transmission lines. The natural period of an overhead line, as given by formula (78) is 1 186000 in which L is now seen to be the length in miles of a quarter wave transmission line, although it was not previously pointed out that the constant 186,000 in formula (78) actually represents the veloc- ity of light (or of an electric impulse) in miles per second. Without attempting to explain or analyze the properties peculiar to a quarter wave length transmission line, it may be said that these are largely due to the fact of the quarter wave displacement providing the charging current for the line, and so leaving the generators to supply the load and losses. The induc- tive pressure drop and the charging current are, in effect, wiped out by the peculiar overlapping of the travelling waves of energy. The power factor of a line specially designed to make use of this peculiarity would, therefore, be very nearly 100 per cent, at all loads, and the regulation, even if the load were inductive, might be surprisingly good. With distorted waves, and complications due to limited length of line, branch circuits, and other causes, it is usually desirable to avoid the conditions of resonance in practice. As an example, consider a line 200 miles long : what frequency will cause the quarter wave effect? This is the frequency which causes the conditions at the sending end to be repeated at the 170 ELECTRIC POWER TRANSMISSION receiving end exactly one quarter of a period later, and by formula (78) we have / = 4X2QO = 233 cycles per second which is the lowest frequency at which free oscillations can occur on a transmission line of this length. It corresponds to the third harmonic of a wave of fundamental frequency 77.7, which should therefore be avoided; but either 60 or 25 cycles will be satis- factory on a line of this length. 86. Reflection of Travelling Waves. Imagine a three-phase transmission line arranged as shown in Fig. 64. Here G is a three-phase generator from which a voltage e as measured be- tween line (1) and ground (or neutral point of star connection) is suddenly impressed for a very short space of time, by closing and then immediately opening the three-pole switch S. We have here the case of a "wave pulse" travelling along the transmission line with the velocity of light. Let us consider what happens when it reaches the end of the line, for the three conditions, C d (3) (2) FIG. 64. Diagram illustrating "wave pulse" travelling along a three-phase transmission line. 1. Line open (/?=<) 2. Line short-circuited (R = 0) 3. Line closed through a non-inductive resistance of value R (line to neutral). Suppose that the operations of closing and opening the switch have been performed so quickly that we have a rectangular wave pulse of which the length ab depends upon the time during which the switch was closed. The current i flows in the wire between the points a and 6, but neither forward of 6 nor backward of a. It carries with it the system of magnetic and dielectric flux lines in the space comprised between the planes ac and bd; the complete energy pulsation being supposed to move away from G with the Velocity of light. PROTECTION AGAINST LIGHTNING 171 Assuming the resistance of the conductors to be negligible, the relation between i and e is given by formula (7 1). 1 To satisfy Case (1), the current arriving at the open end of the line must, at every instant, be equal to the current leaving this point; and since R = oo (i.e., since the circuit is open) i + i' = 0; the symbol i' standing for the " reflected" current. In order to reverse the current and start the wave pulse back toward the generator end of the line, we must conceive of a "piling up'' of the e.m.f. at the open end of the line sufficient to force the reflected current i' against the incoming current i. But that is exactly what must necessarily occur because when half of the wave pulse of current has been reflected, the resultant current in the line will be zero, and since no energy has been lost, it must all be in the electrostatic field. Apart from the relation i + *' = 0, at the open end of the Line, the equations for the outgoing wave, and --- F i r Vc for the reflected wave, must be satisfied; whence it follows that e = e' when i = i'. In other words, during the period of reversal of the current wave, at all portions of the line where i = i', the voltage must be 2e. On arrival at the point where the reversal of the wave pulse is complete, the voltage will again have its original value of e volts. Case 2. With a short-circuit at the end of the line, complete reflection of the wave pulse will also occur, but it will now be the voltage instead of the current that is reversed. Since there 1 This relation is not obviously true of a travelling wave or impulse; but in order to avoid devoting a disproportionate amount of space to this sub- ject, it is necessary to refer the reader to other sources of information, should he find this brief treatment inadequate to his needs. The most practical and the clearest explanation of the peculiarities of travelling waves, known to the writer, will be' found in Franklin and MacNutt's "Advanced Electricity and Magnetism." These authors present the subject in a manner that will satisfy the requirements of most electrical engineers far better than the many highly mathematical writings on the same subject by other authors. 172 ELECTRIC POWER TRANSMISSION is a short-circuit at the end of the line, the volts must be zero when the wave pulse is in process of reversal, and in order to maintain the total energy constant (a necessary condition) the current where the overlapping occurs must be 2i. When reversal is complete, the current will be * as before that is to say, of the same amount and direction; but the voltage e will be of opposite sign because the wave pulse of energy is now travelling along the line in the reverse direction. Case 3. Comparing the two preceding extreme cases, it is evident that since e r = e and i' = i when # = oo, while e' = e and i' = i when R = 0, there must be a particular intermediate value of R which will absorb the wave pulse and prevent reflection. When R is made equal to ~-^> reflection cannot occur because both voltage and current waves will enter the resistance un- changed. Using symbols, we may say that for any value of R, if i is the outgoing current and i' is the reflected current, 1 the balance which enters the resistance is i r = (i + i'}. The volts at the end of the line must therefore be (e + e'} = R (i + i'} and since there is to be no reflection, both e' and i' must be zero; whence Although the cases considered have been chosen merely to explain general principles, and do not exactly represent conditions likely to arise in practice, they should nevertheless be helpful in giving some indication of when troublesome surges or oscilla- tions are likely to occur. Instead of a detached "wave pulse" travelling along a line, we must usually think of a "wave train" of harmonic functions of gradually decreasing amplitude travel- ling along the line in both directions from the point where the dis- turbance occurs. Arriving at the ends of the line, or at points where branch circuits or transformers are connected, these travelling waves of energy may be totally or partially reflected. The reflected waves meeting the outgoing waves may lead to considerable magnification of the original trouble. 1 Both i' and e' (the "reflected" current and voltage) may be positive or negative quantities depending upon their direction relatively to the direction of these components of the original energy wave pulse. PROTECTION AGAINST LIGHTNING 173 Standing Waves. If the conductor resistance is so small as to be negligible, the oscillations of energy may be thought of as a wave train of harmonic functions of constant amplitude travelling to the end of the line which we shall suppose to be unloaded (i. e. , open) . The waves are therefore reflected in the manner pre- viously explained under Case (1), and the reflected waves, meeting the outgoing waves, produce nodes, or points of zero potential, and antinodes where the voltage is exactly double the maximum voltage of the original disturbance. At these points of double voltage the current must be zero since the total energy remains constant and the nodes of the current waves therefore occur at the antinodes of the voltage waves. The com- bination of the outgoing and reflected waves is thus seen to produce standing waves which remain stationary in position although varying in amplitude. The resistance of the conductors prevents this simple mode of oscillation being exactly realized on a practical transmission line. 87. Line Disturbances Caused by Switching Operations. It is hardly necessary to add anything to what has already been said, in order to emphasize the possible danger of suddenly switching a source of electrical energy on or off a long trans- mission line. Unfortunately the calculations of the probable surges or oscillations are not easily made, and moreover accurate data concerning the characteristics of the various circuits and apparatus connected to the system are rarely available. It follows that the engineer cannot predetermine accurately what will happen under the different probable or possible conditions of operation; but a general understanding of the principles underlying the creation of energy surges in a system of electric conductors will enable him to avoid obvious mistakes in the design and operation of a particular transmission scheme. There is frequently danger of abnormally high voltages due to surges at the points where there is a change in the constants of the circuit. Thus, if a transformer is connected across the ends of a long overhead transmission line, there will be a rise of pressure when a travelling wave arrives at this point because the surge impedance I . / ) of the transformer winding may be between 2000 and 4000 ohms, which is very much higher than that of the line itself (about 400 ohms as previously explained). For this 174 ELECTRIC POWER TRANSMISSION reason the end turns of the transformer primaries should be specially insulated to withstand much higher voltages between turns than the remainder of the winding. In the case of a change from underground to overhead trans- mission, a surge originating in the cable will produce a rise in pressure at the junction with the overhead line, while the con- trary will occur (i.e., the voltage will be reduced) if the surge is originated in the overhead line and passes into the cable system of which the surge impedance will always be smaller than that of the overhead transmission. With the good insulation provided on modern high voltage systems, it is doubtful if the interruption of the current by opening switches under load is likely to cause serious voltage disturbances, except in the case of air-break switches where a long arc may be formed and suddenly interrupted as for in- stance by a draught of air when the current is of considerable value. Oil-break switches almost invariably open the circuit at the instant when the current is passing through zero value. 88. Lightning. The foregoing considerations do not take into account the effects of lightning, either by direct stroke or by induction, because in such cases a pressure from an outside source is impressed upon the circuit, and the potential of these atmospheric charges may be tens of times greater than any surge voltage due to a redistribution of the energy stored in the circuit itself. Although our knowledge of lightning phenomena is still far from complete, it is generally agreed that a single stroke of lightning is of short duration, frequently not exceeding the one- thousandth part of a second. If an overhead conductor re- ceives a direct stroke of lightning, the potential value of the lightning charge is generally so enormously in excess of the working pressure on the conductors that the lightning leaps over the insulators down the pole to ground. Any charge on the line, which is not sufficiently high in potential above ground to jump over the insulators, will travel along the line in both directions until it is grounded through a lightning arrester or dis- sipated as I 2 R losses in the conductors. The frequency of such travelling waves will depend upon the natural frequency of the line, and may be of the order of 1000 to 5000 cycles per second. If the resistance of an arrester or the path through which a dis- charge occurs, were zero, the current passing would be a maxi- PROTECTION AGAINST LIGHTNING 175 mum. If C is the capacity in farads, and L the inductance, in henrys, of unit length of line, then*/- is the surge impedance of the circuit; and the maximum possible value of the current will be 7 max . = E -5- -/-, where E is the impressed voltage, which may be considered as something less than the pressure which will cause a flash-over at the insulators. The intense concentration of lightning disturbances is the cause of the difficulties experienced in protecting transmission lines by means of lightning arresters; experience tends to show that an arrester does not adequately protect apparatus at a greater distance than 500 ft., yet it is unusual to find arresters on a transmission line at closer intervals than 2000 ft. Disturbances are most likely to occur on exposed heights, and on open wet lowlands; special attention should therefore be paid to lightning protection at such places. Although the quantity of electricty in a lightning flash may not be very great, the short duration of the flash accounts for currents which are probably of the order of 20,000 to 50,000 amperes. Apart from the effects of atmospheric electricity, it is necessary to guard against the abnormal pressure rises that will occur on long transmission lines through any cause, such as switching operations, or an intermittent "ground." Over-voltages up to 40 per cent, in excess of the normal line voltage can be produced by switching in a long line. High-frequency impulses or surges are set up, which, in the special case of an arcing ground, may give rise to a destructive series of surges, a state of things which will continue until the fault is removed. An arrester which may be suitable for dealing with transitory lightning effects may be quite inadequate to dissipate the charges built up by such continual surges. 89. Protection of Overhead Systems against Direct Lightning Strokes and Sudden Accumulations of High Potential Static Charges. Under this heading the ordinary lightning rod and grounded guard wire will be briefly dealt with. If no guard wire is used, lightning rods should be provided at intervals along the line. They may be fixed to every pole or tower, but, in any case, they should not be spaced farther apart than 300 to 400 ft. un- less the spacing of the supporting poles or towers has to be greater than this, for economic reasons. It is especially important to 176 ELECTRIC POWER TRANSMISSION provide them on the poles or towers in exposed positions such as hill tops. They should project from 3 to 6 ft. or more above the topmost wire. A convenient form of lightning rod is a length of galvanized angle iron bolted to pole top or forming an extension to the structure of a steel tower. Long lines have been worked satisfactorily for extended periods without lightning rods or guard wires, but these are extra high pressure transmis- sions which, on account of the better insulation throughout, are always less liable to trouble from lightning than the lower voltage systems. Although engineers are still divided in opinion as to the value of the protection afforded by overhead grounded guard wires, carried the whole length of the line above the conductors, it is now generally recognized that this method of protection is efficient. The objections to the guard wire are the additional cost and the possibility of the grounded wire breaking, and falling across the conductors below, thus causing an interruption to continuous working. Trouble due to this cause is, however, exceedingly rare. It has been suggested that the guard wire or wires should be of the same material as the conductors, in order that the "life" of all the wires may be the same; but there are other considera- tions in favor of using a galvanized stranded steel cable for the guard wire. This may be the ordinary cable, ^{ Q to J{g i n - i n diameter, as used for guying poles; but, where great strength is required, the Siemens-Martin steel cable, with or without hemp core, is preferable. Bessemer steel wire has not been found satis- factory for this purpose. In the case of the "flexible" steel tower type of line, a strong steel guard wire joining the tops of the towers, adds greatly to the strength and stability of the line, and may even, on long lines, save its cost, by allowing the use of lighter structures and fewer intermediate (dead-ending) towers. In regard to the position of overhead guard wires relatively to the conductors, it is obvious that a number of grounded wires surrounding the conductors will afford better protection than a single wire above the conductors; and two guard wires are some- times provided; but the additional cost is rarely justified. Per- fect protection cannot be obtained even with two guard wires, and cases have been reported of lightning missing the grounded wire and striking a conductor situated immediately below. The best position for a single guard wire placed above the con- PROTECTION AGAINST LIGHTNING 177 ductors is, according to Dr. Steinmetz, 1 such that all the current carrying wires are included within an angle of 60 degrees below the guard wire. Additional wires can be installed in exposed positions, such as the summit, or very near the summit, of a range of hills, or by the shores of lakes or seas where the prevailing winds come over the water. In such positions, an additional guard wire on the side of the conductors may be useful. The guard wire should preferably be grounded at every pole, or at least every 500 ft. 90. Protection of Insulators from Power Arcs. As a special means of protecting insulators from the flash-over caused by FIG. 65. Arcing ring on pin type insulator. lightning, or the power arc following a high potential discharge, the "arcing rings" first introduced by Mr. L. C. Nicholson, may be mentioned. These rings, which are grounded, are placed in such a position as to take the arc, and hold it at a sufficient dis- tance from the porcelain of the insulator to prevent cracking or breakage by heat. The illustrations, Figs. 65 and 66, show the arrangement of the grounded arcing rings attached to stand- ard types of insulator made by the Locke Insulator Manufactur- ing Company. It is not claimed that these rings will protect an 1 Discussion of the Committee on "Lightning Protection," of the National Electric Light Association, May, 1908. 12 178 ELECTRIC POWER TRANSMISSION insulator against a direct lighting stroke; but their utility on high-pressure lines transmitting large amounts of power has been proved without doubt. Although a pair of metal rings, one near the top and one near the bottom of the insulator will probably afford the best pro- tection, the arrangement of arcing rods shown in Fig. 67 will prove almost as effective; the object being not only to provide a path for the high-voltage flash-over which shall keep the arc away from the porcelain, but also to prevent puncture of the insulator due to concentration of potential at the points of , Arcing Ring No.2838 Ring No.2839 Stanislaus Clamp Plan of Ring No.2839 FIG. 66. Arcing rings on suspension type insulator. attachment to the metal fittings. The arcing horns as shown in Fig. 67 are used on the insulators of the 66,000 volt lines of the Peninsular Power Gompany, of Iron Mountain, Mich., de- scribed by Mr. Max. H. Collbohm in the Electrical World of April 18, 1914. 91. Methods of Grounding. The ground wire from lightning rod, guard wire, or arrestor, on high-tension transmission circuits, should be as short and as straight as possible; it need not be of very low resistance; 1 but small reactance is of first importance. 1 The question of resistances in series with spark gap arresters will be taken up in Article 93. PROTECTION AGAINST LIGHTNING 179 The ground plate should have a large surface but the material is of little importance, except that it is not wise to bury aluminum wires in the ground, because of possible electrolytic action. Galvanized iron is a good material. If the ground contact is made with one or more iron pipes buried or driven into the ground, these pipes may be from 1 to 1^ in. in diameter, and a good connection should be made to the top of the pipe, as the inductive effect of an iron tube surrounding the ground wire might be considerable if a connection were made only at the bottom of the pipe. One or more pipes 8 to 10 ft. long, driven into the ground with 6-in. to 12-in. pro- jecting above, will generally be found more effective than buried plates. A very low resistance ground is not essential on a high- tension system, and, generally speaking, the special forms of ground plate made of perforated copper, designed to hold, or to be in contact with, crushed charcoal, are unnecessary. If a plate is used, this should be not less than 12 in. square, but need not be larger than 18 in. square; it maybe of galva- nized iron the charging current per conductor, on the sine- v3 wave assumption, is I c = 2irf^= C e X 10- 6 X L amperes. (86) v 3 Two measurements of capacity can readily be made on the fin- ished cable: (a) between one core (1) and the two remaining cores and the lead sheath, all connected together (2, 3, ). (6) between any two cores, as (1) and (2). FIG. 81. Diagrammatic representation of electrostatic capacities in three-core cable. In terms of the imaginary capacities C a and C n of Fig. 81, we have: Capacity (a) = C, + -~*-T~ = C, + \ C n (87) 1 , 1 o C~n 2C~ n and Capacity (6) = i (C, + C,) (88) There is no constant ratio between wire to wire capacity and the wire to sheath capacity; but, generally speaking, the former (6) is from 57 to 68 per cent, of the latter (a). The most usual 220 ELECTRIC POWER TRANSMISSION value is 60 per cent., whence, by equating (88) with 0.6 times (87), it follows that C 3 = C n . The standard test for electrostatic capacity is between one core and the two remaining cores grounded to the sheath, namely (a) as expressed by formula (87), and since the total capacity (C e ) per core to neutral is obviously C e = C 8 + C n (89) it follows that the approximate ratio of these two capacities is, Effective equivalent capacity, core to neutral (C) Measured capacity, one core to the remaining cores and sheath (a) _ (1 + 1) X 3 _ 3 + 2 The measured capacity (a) for 3-core paper-insulated cables designed for a working pressure of 10,000 volts will be about 0.4 microfarad per mile for a 3 J^-o83 in. cable, and 0.26 for a 1 %-058 in. cable. With shaped instead of circular cores, the capacity is slightly greater, being from 1.08 to 1.1 times the capacity with circular cores of the same cross-section. Calculation of Capacity of Three-core Cables. Although meas- ured values of cable capacities are usually obtainable from manufacturers, it may be necessary to determine approximately the capacity of a cable for a special purpose before it has been made. It would seem at first sight from Fig. 81 and the fact that C s is found to be approximately equal to C n , that C e would be obtained by merely doubling the capacity to neutral as calcu- lated by the usual transmission line formulas (see Article 52, Chapter IV) . It should be observed, however, that the proximity of the sheath will modify the distribution of the electrostatic flux between core and core, and the imaginary capacity C n of Fig. 81 is not what it would be if the lead sheath were removed and re- placed by a considerable extra thickness of insulation. It is found in practice that the capacity of a three-core cable with shaped cores, having the same thickness of insulation between core and core as between core and sheath, is about the same as the capacity of a single-core cable having the same conductor cross-section and the same thickness of insulation between core and sheath. On this assumption, we can use formula (80) of Article 109 for predetermining the probable capacity of a three-core cable. The value of R in formula (80) is taken as the radius of the TRANSMISSION BY UNDERGROUND CABLES 221 conductor plus the thickness of insulation between cores, or between core and sheath. If these thicknesses are not exactly the same, the mean of the two thicknesses is taken. If the con- ductor is not circular in cross-section, the dimension r in the for- mula is the radius of a circular core of the same cross-section as the actual conductor. As an example, let the diameter of each core of a three-core paper-insulated cable be 0.58 in., with insulation between cores 0.38 in. thick, and between core and sheath, 0.30 in. thick. Assuming the specific inductive capacity of the insulation to be 3.3, we have, k = 3.3 r = 0.29 and fi = 0.29 +- - = 0.63 whence, .. , . 0.0388 X 3.3 _ 00 . , Capacity (a) = -- =0.38 microfarad per mile. log (029) The approximate value of the capacity to neutral, for shaped cores, will therefore be, C e = 0.38 X 1.2 = 0.456 and for cores of circular cross-section, C e = -r x- = 0.422 microfarad per mile. 1.08 114. Example of Design of Single-phase Concentric E.H.T. Power Cable. Let the working pressure'be 100 k.v. (alternating) between the inner and outer conductors. The further assump- tion will be made that the maximum stress must not exceed 40 k.v. (r.m.s.) per centimeter. This is a low value, and 50 k.v. would probably be permissible; but the lower figure has been chosen in order to provide a large factor of safety and keep the dielectric loss (and heating) in the neighborhood of its lowest practical limit. Consider first the case of a cable without either "voltage" or "capacity" grading. By formula (83) E 100 r = G=10 =2 - 0m - If a solid core of stranded cable were used, this would correspond 222 ELECTRIC POWER TRANSMISSION to a cross-section of about 2% sq. in. which would certainly be in excess of the requirements, and a hollow core, constructed as shown in Fig. 80, should be adopted. Solving for R in formula (61), we have, 100 " 40 X 2.5 whence ^ = 2.718, and R = 6.8 cm. The dimensions of the cable, in inches, would be approximately : Diameter over outer conductor = 5.56 in. Diameter over lead sheath = 5.93 in. Diameter over armor and jute = 6.6 in. Consider now the alternative of intersheath- or voltage-grading. If a pressure of 50 k.v. is maintained on each side of a single metallic intersheath, the radius r will now be, r- f = 1.25 cm. To calculate the radius over the insulation between the inter- sheath and the inner conductor, we have 1 / R \ 50 gf U.25/ ~ 40 X 1 u.zo/ ^u A i.zo whence R = 2.718 X 1.25 = 3.4 cm. The radial thickness of the lead intersheath would be about 0.05 in., or 0.127 cm., whence the outside radius of the inter- sheath is r' = 3.4 + 0.127 = 3.527 cm. Considering this as the core of a cable with 50 k.v. across the total thickness of insula- tion, and the same maximum voltage gradient as before, we have for the radius over the insulation, Kn 40^527 - ' 354 whence 2~ = 1-425 and R' = 5.02 cm. The approximate dimensions of this cable, in inches, would be as follows: Diameter over outer conductor = 4.08 in. Diameter over lead sheath = 4.45 in. Diameter over armor and jute = 5.12 in. TRANSMISSION BY UNDERGROUND CABLES 223 In order to realize the advantage of this method of grading large high-voltage cables, these figures should be compared with those previously calculated for the cable without voltage grading. It will be interesting to calculate the charging currents per mile of this cable. Assuming the specific inductive capacity of the paper insulation to be k = 3.3 as given on page 213, the ca- pacity per mile between core and intersheath, by formula (80), is C m = -j^ 2 718 = 0.295 microfarad. If the frequency is 25 cycles per second, the charging current, on the sine-wave assumption, will be I c = 27T X 25 X 0.295 X 50000 X lO" 6 = 2.32 amp. Similarly, between the intersheath and the outer conductor, we have _ 0.0388 X 3.3 _ Cm ~ log 1.425 and coo I c = 2.32 X ^i = 6.55 amp. ZvO The difference between these two values of capacity current must obviously be carried by the intersheath, and for every mile of cable the values of the charging current would be as follows: In the core 2.32 amp. In the intersheath . . . 4.23 amp. In the outer conductor . 6.55 amp. Thus, if the cable were 10 miles long, the current in the inter- sheath if fed from one end only would be 42.3 amperes, which might be excessive. This is a point which must not be overlooked in the design and installation of intersheath cables. 115. Losses in Underground Cables. In addition to the PR losses in the conductors, which can easily be calculated, some loss occurs in the dielectric of an underground cable. The ohmic resistance of the insulation being very high, the losses, when a high-tension cable is used on a continuous current circuit, are very small; but with alternating currents there is a further loss due to dielectric hysteresis, which is proportional to the frequency of alternation of the electrostatic field. The total charging current in a cable may be considered as made up of two components, one being the true capacity current of which the phase is exactly 90 degrees in advance of the im- pressed e.m.f., the other being the "energy" component in 224 ELECTRIC POWER TRANSMISSION phase with the e.m.f. This last component is relatively small, being due to what is known as dielectric hysteresis and not to the ohmic resistance of the dielectric, the effect of which is usually negligible. The dielectric loss is equal to the product of the voltage and the "energy" or in-phase component of the charging current. Thus Watts lost = e.m.f. X charging current X cos^> (91) where cos

{ Q of the ultimate stress of the wire. 1 Taken from the National Electric Safely Code, second edition, Nov. 15, 1916. MECHANICAL PRINCIPLES AND CALCULATIONS 273 132. Swaying of Wires in Strong Winds. If a transmission line is well designed and constructed, all the wires of one span will generally be found to swing synchronously in any wind. Under exceptional conditions, however, trouble is liable to occur through wires swinging together, even when all details of design and con- struction have received careful attention. Troubles of this description are more likely to be met with when the spans are long and the sag in the wires necessarily large, and for this reason the spacing between wires must increase with increase of span length, irrespective of voltage considerations. Copper conductors are decidedly less likely to swing out of synchronism than aluminum conductors; not only because the latter have usually to be strung with a greater sag, but also because of the lightness of the material. Aluminum conductors of small sec- tion are easily shaken by sudden gusts of wind, and a little difference in sag will in all probability lead to non-synchronous swinging. It must not be overlooked that wires, after erection, do not always remain equally taut. This may be due to many causes, such as a slight slipping in the ties, straining of insulator pins on cross-arms, unequal ice loading, or local faults in the wires themselves. Again, it has been observed that during snow storms all the wires do not always become coated to an equal extent, and such a want of uniformity in the ice coating may well lead to wires being blown together in a strong wind. On the high-tension transmission system of the Central Colorado Power Company, with spans averaging 730 ft., the lines cross some very exposed positions at the openings to can- yons, and the excessively strong winds that occur at such points have been known to mix up the conductors. It was found necessary to dead-end the line at each tower, guy the towers, and increase the tension in the wires to a point near the elastic limit of the material, steel being used where necessary in lieu of copper. 133. Calculation of Total Stress in Overhead Wires. The formula (101) of Article 125 gives the relation between tension, sag, and weight for a wire strung between supports a known distance apart. Thus, if the tension, P, is known or assumed, the sag can readily be calculated. We shall now consider how this tension can be computed, not only when the wire carries an increased vertical load in the form of ice deposit; but also when the effect of wind blowing across the line increases the total stress. If w r is the total loading in Ib. per foot run of the wire, and w 274 ELECTRIC POWER TRANSMISSION FIG. 98. Vector diagram of forces acting on overhead wires. is the weight in Ib. per foot length of the unloaded wire, then w r = nw, where n is a multiplier which takes account of the extra load on the wire under the most severe weather conditions likely to be encountered in the district where the transmission line is erected. It is usual to assume that the wind pressure acts in a horizontal direction and that the total load on a conductor is the resultant of two forces, one acting vertically downward due to weight of wire together with added weight of sleet or ice, if any, and one acting hori- zontally due to the wind pressure. These forces are indicated in Fig. 98 where Op represents the wind pressure, Ow the weight of the conductor, and ww\ the added weight of ice. The resultant pressure Ow r is equal to Vp 2 + Wi 2 . If the line runs through a country where sleet does not form on the wires the maximum resultant pressure is Or instead of Ow r if the assumed maximum force due to wind is the same in both cases. The diagram Fig. 99 gives values of the multiplier n (i.e., the ratio Or -r- Ow of Fig. 98) corresponding to various wind veloci- ties for standard sizes of solid copper conductors on the assump- tion that there can be no ice formation on the wires, while Fig. 100 gives values of n (i.e., the ratio Ow r + Ow) for copper con- ductors when the weight is increased by a coating of ice 0.5 in. thick with a correspondingly greater wind effect due to the in- creased diameter. The curves of Figs. 101 and 102 give similar relations but for conductors of aluminum instead of copper. The formula used for the calculation of wind pressure in con- nection with these diagrams is p = dV* -5- 4820 where d is the diameter in inches of the conductor or of the ice coating, as the case may be; V is the actual wind velocity in miles per hour, and p is the wind pressure in pounds per foot length of conductor. This is the more correct form of the formula (128) already given. When using the diagrams, it should be noted that the MECHANICAL PRINCIPLES AND CALCULATIONS 275 distances plotted horizontally represent the squares of the wind velocities, and the sizes of the conductors are expressed in equiva- lent B. & S. gauge numbers or in circular mils for the larger sizes. wajonpnoo J9ddoo pjjog jo ja^aajcja The curves of Fig. 99 are correct for solid copper wires: the values of n for stranded conductors would be somewhat greater because of the larger surface presented to the wind for the same 276 ELECTRIC POWER TRANSMISSION 2 2 3 3 3- 2 jojonpuoo ranajraniv popxrea^g P 9reia MECHANICAL PRINCIPLES AND CALCULATIONS 277 vertical loading. The error introduced by using Fig. 99 for stranded cables is not of great practical importance. The curves of Fig. 101 are approximately correct for stranded aluminum conductors; but since the actual diameter will vary with the method of stranding, these charts are intended only for the use of practical engineers who are not interested in mathe- matical niceties. The calculations, on the basis of the assump- tions previously made, are, however, very simple. Thus, if 6 stands for the angle wOr or w\Ow T as the case may be (see Fig. 98), we may write: horizontal loading tan 6 = 7^ ; vertical loading whence 9, and therefore sec 6 can be obtained from trigonomet- . resultant loading rical tables. This last quantity, being the ratio vertical i oadill g is the required factor n when ice loading is not considered. The correction to be made when the vertical load includes ice deposit is simple and obvious. The special case of solid wires without ice coating can be treated as follows: dV 2 Wind pressure per foot length p = ^^Q Vertical load per foot length (copper) w = 3.02d 2 Vertical load per foot length (aluminum), w = Q.92d z (129) 'W~ whence, for solid wires without ice deposit, n (for copper) = ^ 1 + d2 x 21 x 1Q4 (130) I V* n (for aluminum) = ^ 1 + rf2 x 1Q65 x 1Q4 (131) Example: What is the ratio of the resultant load with actual wind velocity of 70 miles per hour, to normal load with wire hanging in still air, in the case of a No. 6 B. & S. copper conductor? The diameter is d = 0.162 in., and by formula (130) (0.162) 2 X 21,200 X 10 4 278 ELECTRIC POWER TRANSMISSION 134. Effect of Temperature Variations on Sag and Stress. Assuming the tension at all parts of the wire to be the same, and equal to P lb., the formula (102) of Article 125 may be written: where I and s are in feet, and w is the weight per foot of the wire. If wind or ice or both result in a loading per lineal foot equal to to nw, it follows that the tension in the loaded wire is now n times as great. If the symbol S be used to denote the stress or tension in the wire per square inch of cross-section, the formula giving the relation between tension and sag may be written, S = kn- (133) $ in which k is constant for a given material; it is equal to g-r where A is the cross-section of the conductor in square inches. The numerical value of k is therefore one-eighth of the weight in pounds of 12 cu. in. of conductor material, or 1.5 times the weight of a cubic inch. The values of k, together with ultimate and working values of the stress S, will be found in the table of physical constants in Article 41 of Chapter IV (p. 75). The length of wire between fixed supports of equal height bears a definite relation to the span and sag. This relation, as already given, is: x - i +'| ! (in) where I is the distance between supports and s the sag at center, both expressed in feet. The increase in length due to stress will be directly propor- tional to the tension per square inch (S) provided the elastic limit of the material is not exceeded. The approximate elastic limit for conductor materials is given in the table of Chapter IV. The formula for elastic stretching, as already given, is X e = X^ (125) where X is the length of wire when S = 0, X e is the elongation due to the stress S, and M is the elastic modulus. The increase of length due to rise of temperature, on the as- sumption that stress remains unaltered, is: X X a X i (134) where a is the temperature-elongation coefficient. MECHANICAL PRINCIPLES AND CALCULATIONS 279 Values of k, a, M and the maximum safe working stress S, taken from the data in Chapter IV, are as follows: For copper k = 0.485; a = 0.0000096; M = 15,000,000; S = 28,000 For aluminum k = 0.146; a = 0.0000128; M = 9,000,000; S = 13,000 For copper-clad steel (40 per cent.) k = 0.447; a = 0.0000067; M = 20,000,000; S = 40,000 For iron wire k = 0.423; a = 0.0000066; M = 25,000,000; S = 26,000 For Siemens-Martin steel k = 0.427; a = 0.0000066; M = 29,000,000; S = 33,000 It is a simple matter to calculate by means of formula (132) or the modified formula (133) the sag corresponding to any span (0, load (nw) and tension (S), and by using either of these formulas, the minimum allowable sag, for the safe limiting tension when the wire is subject to the greatest expected load in the matter of ice and wind pressure, should be determined in the first instance. Having determined the amount of this sag which should be that corresponding to the lowest expected tem- perature the length of the wire in the span can readily be cal- culated by means of formula (111). The calculation of the sags and corresponding tensions at other temperatures and under other conditions of loading is not by any means so simple a matter, because the alteration in the length of the conductor depends not only upon the temperature, but also upon the tension. If the extra load on the conductor due to wind pressure and ice (if any) be removed, the sag will adjust itself until the formula (132) is again satisfied, the weight w being in this case that of the wire only. A further condition is that the length of wire shall be equal to the length as origi- nally calculated for the loaded wire, less the elastic contraction due to the reduction in the tension. If the temperature be now supposed to rise, the length of the wire will increase, but not in di- rect proportion to the temperature rise as indicated by formula (134) because so soon as there is any increase in length leading to an increased sag, the tension in the wire is immediately relieved, and since it is assumed that the elastic limit has not been ex- ceeded there will be a reduction in length which could be cal- culated by formula (125) if the amount by which the tension is re- lieved were known. 280 ELECTRIC POWER TRANSMISSION A mathematical formula which expresses the required length or sag under normal conditions, in terms of the length correspond- ing to minimum temperature and heaviest loading, is very com- plex and difficult of solution. Mr. H. J. Glaubitz 1 has evolved an equation in which the first and third power of the unknown quantity (the deflection or sag) appear simultaneously. The solution of such an equation is tedious and is usually accomplished with the assistance of more or less scientific guesswork. A graphical method, which is probably in more general use, consists in plotting two curves, one showing the relation between sag and tension for the selected span when the wire hangs naturally under its own weight only, and another curve calculated for a definite constant temperature and giving the relation between sag and tension when a wire of definite known length under a known tension is subjected to various assumed changes in the tension. The point where the two curves cross will indicate the required conditions of sag and tension. This process is a lengthy and laborious one and has to be repeated for every assumed change of temperature. Graphical methods of calculating sags or tensions under various conditions of temperature and load are quite accurate enough for practical purposes, and since the material published in this connection is so abundant and varied that there is room for individual choice in the selection of a particular method, or chart, or combination of charts and diagrams, the needs of the engineer who prefers graphical methods to the more lengthy analytical processes, are easily satisfied. For one who is working constantly on the same kind of problem, diagrams or charts are usualy of very great assistance; but if a method involving any but the simplest diagrams is not made use 1 Electrical World, March 25, 1909, p. 731. The reader may also refer to the Electrical World of July 13, 1912, p. 101, where Mr. H. V. Carpenter evolves a formula containing the third and first powers of the unknown quantity, and proposes a chart to assist in arriving at the solution. An excellent article on Sags and Tensions in wire spans from the pen of Dr. Harold Fender appeared on page 604 of the Electrical World of Sept. 28, 1907. Among more recent contributions there is an article by Mr. K. L. Wilkinson in the Electrical World of Feb. 6, 1915, and a paper by Mr. A. T. Arnall on p. 360 of vol. cci (June, 1916) of the Proceedings of the Institution of Civil Engineers (England). Practical suggestions for solving the equa- tions containing the third power together with the first or second power of the unknown quantity are made in these papers. MECHANICAL PRINCIPLES AND CALCULATIONS 281 of very frequently, mistakes are likely to occur, or else the time spent in re-studying the method of procedure leaves the graph- ical treatment without advantage over the more tedious process involving the solution of mathematical equations. The writer has always found charts or diagrams of great assistance in practical engineering work, and in the first edition of this book an accurate method of solving sag-temperature problems by means of two superimposed diagrams was described; but, for the reasons above given and also to avoid the introduc- tion of unnecessary material, the graphical method of procedure is omitted from this edition. The analytical method about to be described does not require the solution of equations con- taining the third and other powers of the unknown quantity: it is accurate, and does not include any difficult mathematical processes. 135. Calculation of Sags and Tensions under any Conditions of Load and Temperature. This method of calculation requires the knowledge of the sag and tension corresponding to one par- ticular temperature. A different sag is then assumed, and the temperature at which this sag will occur is calculated by means of a simple formula. The manner of obtaining the preliminary data will be explained later. It is assumed that the conductor is strung between two fixed supports on the same level, and that the material of the conductor is not strained beyond the elastic limit. The meaning of the symbols used is as follows: I = the length of span, or horizontal distance between points of support, in feet, s = the vertical sag at center of span, in feet, when wire hangs in still air under the influence of its own weight only. P = the tension in the conductor at the lowest point of span, in pounds. S = the stress in the conductor at the lowest point of span, in pounds per square inch of cross-section. X = the length of conductor measured between the two points of support. t = temperature, in degrees Fahrenheit. s c , S c , X c = known values of sag, stress and length cor- responding to a definite temperature t c , when wire 282 ELECTRIC POWER TRANSMISSION hangs in still air under the influence of its own weight only. a = the coefficient of linear expansion of the conductor per degree Fahrenheit. M = the modulus, or coefficient, of elasticity of the con- ductor, being the ratio of stress in pounds per square inch to extension per unit length. w = the weight of conductor in pounds per foot of length. w r = the resultant or total load in pounds, per foot, includ- ing wind pressure and ice (if any). n a multiplier depending on the material of the con- ductor and weather conditions, being the ratio w r /w, when supports are on the same level. k = a constant depending upon the material ofthe con- ductor, being 1.5 times the weight in pounds of a cubic inch of the conductor material. The well-known formula giving the relation between sag, length of span, horizontal load, and tension is: . - (101) The approximate formula for the length of a parabolic curve (which is quite sufficiently accurate for practical purposes) is: x = J + |- 2 (ill) It is assumed that the sag s c , and therefore the corresponding stress S c and length X c are known for the particular temperature l c , which may be fairly high so that another value, s, of the sag, arbitrarily chosen, shall be smaller than s c ] and it is proposed to calculate the temperature t which will correspond to this assumed sag, s. With the reduction in the amount of sag, there must of neces- sity be a reduction in the length X c of the conductor and an in- crease in the tension P c or stress S c . The amount by which the length has decreased is not directly proportional to the reduc- tion in temperature, because the increase in tension causes an elastic elongation of the conductor, and the reduction in length is actually the difference between the amount by which the wire would contract with the lowering of the temperature if the ten- sion were to remain constant, and the amount by which the wire MECHANICAL PRINCIPLES AND CALCULATIONS 283 would be extended, due to the increased tension, if the tempera- ture were to remain constant; although, from a strictly scientific point of view, the argument may be inaccurate. The decrease in length due to temperature reduction is: X c X a X (to - (135) and the increase in length due to additional tension is: &(*%-), M \s I ' Therefore, Xc - X = X c X a X (* c - - jjScQ - l) The lengths X and X c can be eliminated by substituting their values in terms of sag and length of span, as given by formula (111). This leads to the equation: 8s c * - 8s 2 ,_&/! _ i\ Q37) ~ (3* 2 + 8s c 2 )a + aMs This formula is very simple to use, because for a given material and size of wire, it may be written: tc - 1 = ^-^ + ^-1} (138) A2 \S / where Ki, K 2 , and K 3 are constants, the values of which are: K! = 8s c 2 (139) K z = (3Z 2 + 8s t 2 )a (140) . Moreover, since 8s c 2 is always very small in comparison with 3 2 , the constant K 2 may, for nearly all practical purposes, be written : K 2 = 3al* (approx.) (142) This value of K 2 may be used for spans up to 500 ft. if the multiplier n does not exceed 12, and for spans up to 1000 ft. if n does not exceed 6. In the case of longer spans, in which the sag is relatively large, or if a closer approximation is required, the more exact expression (140) should be used. The only unknown quantities in equation (137) or (138) being the sag s 284 ELECTRIC POWER TRANSMISSION and temperature t, it follows that, by inserting any numerical value for s in the equation, the change of temperature, and there- fore the actual temperature t corresponding to the assumed value of the sag, can be readily calculated. In order that this method of calculation may be of practical utility, it is necessary that the sag s c and the stress S c at the particular temperature t c shall be known. The fundamental data on which all line calculations are based must include the limiting or maximum allowable value of the stress and the conditions of maximum loading under the most severe weather conditions. The maximum load per foot being w r and the weight of the unloaded wire being w, it follows that the ratio = n will be greater as the wind conditions, either without ice or combined with a coating of ice on the wires, are the more severe. The wires will generally be subject to the greatest stress at times when strong winds, with or without a coat- ing of ice or sleet, occur at a low temperature, because the low- ness of the temperature alone will account for a considerable increase in the tension. If the extra load on the wire due to wind and ice combined is great in proportion to the weight of the wire, the maximum de- flection will usually occur under winter conditions; but there will be a higher temperature at which the sag of the unloaded wire hanging in still air, subject to its own weight only, will be exactly the same as the deflection under winter conditions when subject to wind pressure and extra load of ice (if the line runs through a district where sleet and ice formation is possible). This tempera- ture, which may be called the critical temperature for the ma- terial of the conductor when the maximum winter loading has been determined, is easily calculated; and its numerical value, together with the known value of the sag under conditions of maximum load, and of the tension corresponding to this sag, may be used in equation (137) or (138) for the known quantities t c , s c and S c . Calculation of Critical Temperature t c Let S m be the stress in the wire under the most severe conditions of load, and t the temperature at which this stress occurs. The tension S c will be equal to S m divided by n, because, at the crit- ical temperature t c , the sag is the same as the maximum deflection of the loaded conductor, but the weight per foot of length is in MECHANICAL PRINCIPLES AND CALCULATIONS 285 the ratio or - (Refer to formula (101), bearing in mind that, for any given size of conductor, the stress S is proportional to the tension P.) With an increase of temperature from t to t c the reduction in stress is and the reduction in length of the wire due to this difference of tension is It is required to calculate the temperature rise (t e to) which will produce an extension exactly equal to this elastic contraction, in order that the length X c of the wire, and consequently the sag s c , shall remain as before. The extension due to temperature rise is X c X a X (t e - t ) and the required equation is : X c X a X (tc - to) = ^ S m (l - i) or ftJ-r'tl - Sm d . to) ~MXaV The curves in Fig. 103 give the relation between the ratio (being the reciprocal of n) and the temperature rise (t c t ) for stranded cables of different materials. The values of M and a, which have been adopted for the purpose of drawing the diagram, are: For hard-drawn copper, M = 15 X 10 6 , a = 9.6 X 10~ 6 For hard-drawn aluminum, M = 9 X 10 6 , a = 1.28 X 1Q- 5 For galvanized steel, M = 25 X 10 6 , a = 6.5 X 1Q- 6 Having determined the critical temperature t c at which it is interesting to note the tension in the wires, if correctly strung, will be the same whatever the length of span, the sag s c can be calculated by the formula (101), or by the more convenient formula : kl 2 s. (144) which can be put in the form knl* o (145) Om 286 ELECTRIC POWER TRANSMISSION In this manner the numerical values of the quantities t c , s c , and S c for use in formula (137) or (138), are obtained. Example. Construct a sag-temperature chart for the use of the construction engineers in the field, based on the following data : Horizontal span = 485 ft. with rigid supports on the same level. v \ Chart f will m hangin to defl maxim Ordina Where iving te ke sag g unloa ction u urn stre es of C P a t M -n mperature rise which Of stranded conductor fcd in still air equal nder conditions of \ \ N s \ < laxiraum stress, Ibs. er sq. inch ^mp.coefficent odulus of elasticity itio of resultant or tal load per foot nder worst conditions ) weight per foot of nloaded wire \ \ s \ s s \ \ \ \ t n t u \ S \ k \ s \ \ > k N \ \ X \ V % ^ \ X..N \ i N% ^ ^s. \ XN <'** 5 \ < ^s^ *V fe \ S C % s K\ V ^ $s^ is \N SN ^s^ s\ \ ^ ^v"N ^s k ^^0\ ^S i^ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Ratio ^-(beins reciprocal of multiplier n ) Fio. 103. Chart for determining "critical temperature." Conductors: stranded aluminum No. 2/0 B. & S. Stress not to exceed S m = 14,000 Ib. per sq. in. with a combined load of 0.5 in. ice coating and 47-mile wind at a temperature t = - 20 F. From Fig. 102, the value of n for V 2 = (47) 2 = 2210 is 8, whence I = - 125 - MECHANICAL PRINCIPLES AND CALCULATIONS 287 From Fig. 103 (t c - to) = 106 whence t e = 106 - 20 = 86 F. this being the "critical temperature" at which the vertical sag of the unloaded conductor hanging in still air will be the same as the maximum deflection of the loaded conductor under the speci- fied conditions. Other required values are S c , ,,..,. By formula (144), S c = 1750, and k = 0.146. 0.146 X (485) 2 . ~I750~ By formula (139), Ki = By formula (142), K 2 = By formula (141), K 3 = 8 X (19.6) 2 = 3070 3 X 485 X 485 X 1.28 X 10~ 8 1750 X 10 5 9 X 10 6 X 1.28 By formula (138), (86 - t) = 3070 - 8s 2 15.2 + 15.2 19.6 Temperature, Degrees Fahrenheit ?sosggggggsi / 180 Size Aluminum No. 2-0 Span =485 Feet p / / / / 190 / / 195 4 >* / Note; Figures on Curve Indicate Required Tension on Dynamometer / '20, xl 10 / / 215 17 1 V 2 18 18H 19 19^ 2 20 Sag at Center of Span (Feet) FIG. 104. Sag-temperature curve for use when stringing wires. By choosing values for s not very different from s c the corre- sponding temperatures are easily calculated. Thus when s = 16^ ft., t = - 15.8 F. when s = 17M ft., t = + 15.4 F. when s = 18M ft., t = + 48.4 F. when s = 203^ ft., t = + 120 F. With the aid of these figures a curve such as Fig. 104 is readily plotted. This curve gives the men in the field all necessary 288 ELECTRIC POWER TRANSMISSION information for the correct stringing of the conductors, whatever may be the temperature when the work is carried out. 136. Tensions in Conductors when Spans are of Different Lengths. It is well to keep the consecutive spans in a trans- mission line as nearly as possible of the same length, because, although it is possible to string the wires so that the tension shall be the same in all spans at the time of stringing or under specified conditions of load and temperature, there will be an unbalancing of the tensions in adjoining spans with every change of tempera- ture. If properly strung, the wires in long and short spans should be subjected to the same maximum tension under the severest conditions of loading, and the condition of equal tensions will repeat itself at the higher temperature previously referred to as the critical temperature when the sag of the unloaded con- ductor is the same as the deflection of the loaded conductor at the lower temperature; but, except when the deflection at center of span remains unaltered, 1 the pull on each side of a supporting insulator will be unbalanced. It is partly for this reason that extra long spans are usually "dead-ended" on guyed poles or strain towers. When calculating sags in spans of different lengths, it is therefore not correct to assume that the sag is always directly proportional to the square of the length of span; because when the wire hangs in still air subject to its own weight only, this propor- tionality exists for no other temperature but the critical tempera- ture as determined for use in the sag-temperature calculations. 137. Tension in Different Sized Wires on the Same Span. It may be questioned whether, having calculated the sag- tempera- ture conditions for a conductor of diameter d\, there is not a short cut by which similar relations can be arrived at for a wire of diameter d z when the length of the span, I, remains unaltered. There does not appear to be a quick way of obtaining the required results; but there is one condition that holds: ir The condition is that the quantity shall remain constant. n ni This expression is derived in the same manner as the formula (143) : there must obviously be some particular value of the wind or ice loading (corre- sponding to the factor n\) which, in conjunction with a rise of temperature (t\ to) will cause the deflection to be the same as when the temperature is t and the (maximum) loading is n times that due to the weight of the wire acting alone. The necessary relation between t\ and i is given by the above formula. MECHANICAL PRINCIPLES AND CALCULATIONS 289 Let n be the load coefficient ( } for conductor di, and n 2 the load coefficient for conductor d z then, since I 2 is assumed constant, it follows from the formula (145) that and s * = s ' x rT* which is true only when the stress () per square inch of cross- section is the same for both sizes of conductor (the material of the conductors being assumed the same). 138. Further Example Illustrating Temperature-sag Calcu- lations. Although the problem may not be of great practical utility, it will serve as a further illustration of the methods of calculation explained in Article 135. Determine the reduction of temperature which mil make the tension in the unloaded conductor the same as in the loaded conductor when the resultant load is n times the weight of the wire. When calculating the "critical temperature," the formula (143) was developed. This enables us to calculate the rise of temperature necessary to make the length of the unloaded con- ductor hanging in still air equal to that of the same conductor at a lower temperature, but with an extra load due to wind or ice, or to both combined. The problem now before us is con- cerned with the stress, which must be the same for the loaded and unloaded conditions. Since the stress remains unaltered, the change in length of the conductor corresponding to the (necessarily) smaller sag is caused by reduction of temperature and not by elastic contraction. If s = maximum deflection under loaded condition, and Si = sag of the unloaded wire when the temperature has fallen t F. and the stress S is again equal to what it was under the loaded condition, then, s Sl= n # S The reduction in length is, \! X a X t = \ - \i 19 290 ELECTRIC POWER TRANSMISSION which resolves itself into 8'(n - 1) ~ 3n 2 X a X I No appreciable error will be introduced, when dealing with spans of moderate length, if the length of wire Xi is replaced by the length of span I. Hence, = 8s*(n* - 1) 3n 2 XaX / 2 which, if it is desired to eliminate s, may be written _ 8(n 2 - 1)W " 3 X a X S 2 (147J It may be observed that, for a given material and limiting tension, the required reduction in temperature is proportional to f n z ]\l 2 (n 2 1) X Z 2 , or t = ~j ' where K is a numerical constant. By using the data for materials previously given and assuming maximum allowable tensions corresponding to S = 28,000 for copper; 13,000 for aluminum; and 25,000 for steel guy wire, the calculated value of K is, For copper K = 12,000 For aluminum K = 38,000 For steel K = 8,250 139. Sag-temperature Calculations with Supports at Different Elevations. We shall consider (1) the case of a small difference of elevation between supports, which will cause the lowest point in the span to be below the level of the lower point of support, as illustrated in Fig. 105; and (2) the case of a line running up a steep incline, which will cause the lower support to be the lowest point in the span (see Fig. 106). The problem will be studied by working out numerical examples. Data for numerical examples: Wire; No. B. & S. solid copper. Diameter of wire, d = 0.325 in. Cross-sectional area of wire, A = 0.083 sq. in. Weight of wire per foot run, w = 0.32 Ib. Distance between points of support (measured on incline), V = 240 ft. Breaking stress (say) 55,000 Ib. per sq. in. In these examples, we shall neglect ice-coating on the wires, MECHANICAL PRINCIPLES AND CALCULATIONS 291 but allow for a high wind velocity (78 miles per hour) giving 15 pounds per square foot of projected surface of the wire, and we shall also adopt the abnormally high factor of safety of 5 as called for in the regulations issued by the British Board of Trade. The maximum allowable stress, at a specified temperature t = 22 F., is therefore s ^55,000 5 = 1 ] ,000 Ib. per sq. in. Maximum tension in wire, P max . = 11,000 X 0.083 = 915 Ib. 325 The wind pressure per foot run of the wire is p = 15 X vs- 12 = 0.406 pounds, and the value of the factor n as defined in Article 133, is V(0.32) 2 + (0.406) ' 0.32 1.58 FIG. 105. Wire hung between supports at different elevations. For the calculation of the critical temperature (see Article 135) we have, M = 15 X 10 6 and a = 9.6 X 10~ 6 , whence, by formula (143), (t c t ) = n c vx i c vxirkfi ( 1 ~ T~58/ 28 9.6X15X10 6 ' and t c = 28 + 22 = 50 F. It is at this temperature that the length of the wire hanging in still air will be the same as the length under the condition of maximum load when the stress is n times as great. It follows that the tension in the wire at a temperature of 50 F. (without wind) will be Poi P\ max. -lO __ Q ,. c = ~ = < x~^ = O/O ID. n l.oo Case (1). Small Difference of Elevation. (See Fig. 105.) For the additional data required, assume: 292 ELECTRIC POWER TRANSMISSION Difference in elevation of points of support, h = 13 ft., whence 1 O sin e = 240 = - 0542 > and cos e = - 9985 > which indicates that the span measured horizontally, being I = l r cos 6, is so nearly equal to I' that no appreciable error will be introduced if we neglect to take account of this difference. In this particular example it is therefore a matter of indifference whether we put I or I' in the formulas. Now choose two or three values of tension in addition to the tension P c = 578 at the "critical" temperature of 50 F., and calculate the corresponding temperatures by the step by step method as followed in the accompanying table. This method of procedure is correct except for the fact that formula (123) used for obtaining item (/) does not include all the terms necessary for the exact calculation of the length of a para- bolic arc. With a difference in level of only 13 feet with supports spaced 240 feet apart, the error is probably negligible; but with a greater difference in elevation between the points of support (as in Case (2) about to be considered) additional terms would have to be included in the equation, thus adding to the time required for the calculations. The maximum deflection (item (c)) of the wire from the straight line AB joining the points of support is always small relatively to the distance I' = AB] and in computing the length of a curve which approximates to a straight line no appreciable difference will be observable whether we consider the curve to be part of a parabola, or catenary, or ellipse, or circle. For our purpose it will be most convenient to calculate the length (X) of the wire by using formula (122) of Article 127, merely substituting s' (item (c)) for s, and I' (see Fig 105) for I. This has been done in the example, Case (1), the results being given under items (/)' and (gY at the end of the accompanying table. The change in length (X X c ) is seen to be the same whichever method of calcu- lation is used, 1 and since the latter method of calculating X is much shorter than the former, it should be adopted when making sag-temperature calculations for spans on an incline. Case (2). Large Difference of Elevation. (See Fig. 106.) For additional data required, assume: 1 The very small difference in the two sets of figures may be attributed to errors in reading the slide rule. MECHANICAL PRINCIPLES AND CALCULATIONS 293 OJ C "3 OS Cc d do d o > ^ ^H : . j : ^ : : g : a -g : i-H 3 ^ -g + i : ! | ^efl a & J , : : fe : . : 9 '- : .a ^ vx vx ft- \w rj - -g 8 (^tc'-fj S^QS '^-ii'o '2* I "^ ' *bfi CO ^ ^ ~*^ ~|" ^ ' "M ' Q- 5 .s flQn-'S '"'d' S > ' '** ' w .!! 1 1 oS ^%: aT o '' * ^ : 2 " ^ i ji'f i| i;7 t r | ?.' i^i ? s i w ff * S ' ^-^ j3 ii Q** . ^O t> ^3 02 X * O '^ " O -|- rf 8, : S : ^ ^ C | < ^^ : ' < S 1 1-! y.| 1 | ifelvs % I "^SiBfl^ S | V ~v'ia, >V i^ + 1 : I 5 1 o o o 5 5 9 ** ** Iil s ^* ffi m '> ~ whence t c = 35 + 22 = 57 F. We are now in a position to proceed as in Case (1) (alternative method), or if preferred we may use the formulas of Article 134 which will give exactly the same results. Let the assumed values of tension, in addition to the "critical" value P c = 494 lb., be PI = 600 lb., and P 2 = 400 Ib. The corresponding maximum deflections from the straight line, as calculated by formula (148) are, s' e = 3.79 ft. s'i = 3.12ft. s' t = 4.68 ft. By formula (139), KI = 8 (3.79) 2 = 115 By formula (142), K 2 = 3 X 9.6 X 1Q- 6 .X(240) 2 = 1.66 494 By formula (141), K 3 = aQ83 x 9 . 6 x 15 = 41.5 By formula (138), putting s = 3.12 ft., we have, (7 n H5-8(3.12) 2 /3.79 (57 ~' l): ~L66~ 4l ' 5 (^T2 whence ti = 24.7 F. Similarly, when s = 4.68 * 2 = 101 F. The curve marked (2) in Fig. 108 has been plotted from these results, while curve (1) refers to the previously calculated Case (I). The correction for n, and therefore for the "critical" temperature t c , need not be made when the slope is small. It is only when the difference between the horizontal spacing (I) and the actual distance between supports (I') is appreciable that the correction need be made. The procedure here recommended MECHANICAL PRINCIPLES AND CALCULATIONS 297 for constructing sag-temperature curves for lines on an incline consists in replacing the actual span by an "equivalent" span with supports on the same level. This equivalent span may be defined as a horizontal span of the same length (l r ) measured between points of support as the actual span, the loading being w cos 6 Ib. per foot run, where w is the weight per foot of the wire, and Q is the angle between the direction of the line and the horizontal. 130 Case(2) pan on considerable /400' Cased) Span Horizontal on Slight Incline Note The figures on the curves indicate the tension in tha wire in Ibs. 3.5 4 4.5 5 5.5 6 Sag, 01 Maximum Deflection from Straight Line Feet 6.5 FIG. 108. Sag-temperature curves illustrating numerical example. The dotted curve marked (3) in Fig. 108 is plotted from the same data as curve (1) except that the effect due to elasticity of the wire has been neglected (item (h) in the table on page 293). The omission is often permissible, especially on long spans of aluminum wire; but, on comparatively short spans of copper wire, as in the present instance, it is seen to give entirely inaccu- rate results. The point is mentioned here because of the peculi- arity that in England where the stringent regulations involving large factors of safety necessitate uneconomical construction with short spans it is by no means uncommon to neglect the elasticity of the wire; while in America where the longer spans 298 ELECTRIC POWER TRANSMISSION often cause the inclusion of this item to become an unnecessary refinement it is customary to take account of the changes in length due to variation of load. 140. Length of Spans. Conductor Materials Copper ; Alumi- num ; Iron. When the wires of a transmission line are supported on single wood poles, the span may be anything from 120 to 250 or even 300 ft. ; but extra long poles, specially selected to withstand the greater stresses, are required when the span is appreciably in excess of 200 ft. On steel tower lines, very much greater spans may be used. The length of span is determined not only by strength considera- tions, but also by considerations of economy (see Articles 15 and 16 in Chapter III). The requirements in the matter of supporting poles or structures (which depend largely on length of span) will be referred to in the following chapter. Whether copper or aluminum should be used on a given trans- mission line cannot be determined on general principles. Con- ductor materials were discussed in Chapter IV, and, apart from the physical properties of these materials, the relative cost, which is a variable quantity, must be taken account of when deciding upon the material best suited for the work. Generally speaking, the deflections or sags of aluminum con- ductors on spans of moderate length will be about 30 per cent: or 35 per cent, greater than with copper conductors. The difference will be more marked with the smaller sizes of wires and the shorter spans. With extra long spans in the neighborhood of 1000 ft., it will be found that the maximum sag of aluminum and copper conductors will be about the same, if storm and abnormal winter conditions are neglected; that is to say when the factor n (defined in Article 134) is unity. Under this condition it will even be found that copper has a greater sag than aluminum on the very long spans. The reason for this is that the greater temperature elon- gation of aluminum is inappreciable in the case of long spans with necessarily large sags, while it is an important factor in the com- parison of the two metals when the spans are short. The above statement is made rather as being of scientific interest than of practical utility, because, under storm conditions (when n has a large value), aluminum will be found to be an unsatisfactory material to use on long spans. Although the tension in a conductor can always be kept reason- ably small by allowing sufficient sag, it is obvious that a large sag MECHANICAL PRINCIPLES AND CALCULATIONS 299 involves higher and more costly supporting structures if the clear- ance between ground and lowest point of the conductor is to remain the same. When crossing open country, the clearance above ground need not be very great; but in crossing roads, a clearance of 21 feet should be allowed, and over foot paths the clearance should never be less than 15 ft. For mechanical rea- sons, it is a general rule that no conductor for the transmission of electric energy shall have a lower ultimate strength than a No. 6 B. & S. gauge hard-drawn copper wire. Although it is convenient in sag calculations to assume rigid supports at each end of the span, the deflection, under heavy load, of pole or steel mast as used on the shorter spans, may be re- garded to some extent as a factor of safety. Refinements of calculation are out of place when figuring on short span lines : the tendency is to string wires of short-span transmission lines too slack rather than too tight. On spans not exceeding 150 ft., if the wires are strung at comparatively low temperatures, it is almost impossible to draw them up too tight. The writer is of the opinion of Mr. H. W. Buck and one or two other eminent and experienced engineers, who hold that most transmission lines are too slack. The fact that the natural stretch which takes place in all overhead conductors during the few months following erection, increases the sag which is usually excessive in the first instance, is frequently overlooked; or, if it is recognized, the additional sag is erroneously supposed to increase the factor of safety. It is not good engineering prac- tice to have the wires hanging in festoons between the supports, and the danger of short circuits resulting from slack conductors being thrown together in stormy weather is unquestionably greater than the risk of breakages on a well-constructed line with wires strung tightly between towers of sufficient strength to withstand the maximum loads that they may be subjected to under the worst weather conditions. EXAMPLES OF EXTRA LONG SPANS Where power lines cross rivers or other navigable waterways, exceptionally long single spans are sometimes necessary. The longest in the world crosses the St. Lawrence river about 20 miles from Three Rivers, Quebec, and measures 4800 feet. The ends of the span are attached to steel towers 350 feet high. Construc- tion details will be found in the paper by Mr. S. Svenningson, 300 ELECTRIC POWER TRANSMISSION Proc. A. I. E. E., Nov., 1918. Another long single span is the Missouri river crossing of the Mississippi River Power Co. which is 3182 feet long: the conductors consist of copper wires of a total cross-section of 0.162 sq. in. laid on a central steel core of 0.275 sq. in. cross-section. The Pacific Light and Power Corporation have a 2871-foot span at Sunland, Cal. : in this case the conductors are of aluminum wire of 0.475 sq. in. total cross-section laid on a central steel core of 0.062 sq. in. cross-section. The Great Western Power Co. at Oakland, Cal., have a 2740-ft. span over the navigable waterway: it consists of copper-clad steel conductors, of 0.132 sq. in. cross-section. The Niagara River crossing of the Canadian Niagara Power Co. is 2192 ft. long: it consists of copper-clad steel conductors 0.155 sq. in. in cross-section. The longest existing span using aluminum cables without steel core is at Piedra, Cal., on the system of the San Joaquin Light and Power Co.: it is 1700 feet long; the conductor cross-section being 0.132 sq. in. The possibilities of iron wire as a conducting material for over- head power lines was considered in Articles 46 to 49 of Chapter IV. It is only when the price of copper and aluminum is abnor- mally high that iron as a conductor material need be seriously considered; but in certain cases such as short lines transmitting small amounts of energy the saving resulting from the use of iron wire may be considerable. 1 This is due partly to the fact that, with conductors of % in. or % 6 in. galvanized steel strand cable, the spans may safely be made longer than when the equiva- lent copper wire is used. Thus, for the purpose of transmitting from 50 to 80 kw. a distance of 20 to 30 miles by a single three- phase line, the average span on a straight run across country might be 300 feet with 30-ft. poles, 500 ft. with 35-ft. poles, or (preferably) 600 to 700 feet with 40- to 45-ft. poles; and the cost of such a line excluding the cost of wire 'might with care and proper supervision be no more than two-thirds of the figure indicated by the lower curve of Fig. 18 (Chapter III). Wood poles will usually be found satisfactory for this cheap 1 Useful mechanical data referring to iron and steel conductors will be found in an article by Messrs. Oakes and Sahm in the Electrical World of Aug. 10, 1918, vol. Ixxii, p. 249. MECHANICAL PRINCIPLES AND CALCULATIONS 301 type of construction, but with the longer spans it is probable that steel poles or masts handled and erected in one piece, with or without concrete settings, will frequently prove more economical than wood. Double-galvanized wire or cable should be used. For very long spans it may be advisable to use special high-grade steel in order to avoid excessive sag and spacing be- tween wires, and to pay particular at- tention to the guying of corner poles. 141. Factors of Safety. Joints and Ties. The factors of safety usually adopted when calculating the permis- sible tension in overhead wires have already been referred to in Article 131, and little remains to be added here. In America, the factor of safety for conductors is about 2 : this means that, under the worst assumed conditions of loading, the material of the conductor may be stressed to very nearly its elastic limit. It is usual to assume a wind velocity of 70 miles per hour com- bined with a sleet deposit H in. thick at a temperature of F. , For guy wires, a factor of safety of 3 to 3>^ is generally allowed. In Great Britain, where higher fac- tors of safety are used, the Board of Trade calls for a maximum tension not exceeding one-fifth of the breaking load, on the assumption of a temperature of 22 F. and a wind velocity correspond- ing to a pressure of 15 Ib. per square foot of the projected surface of the wire. Possible accumulations of snow and ice are ignored. On the Continent of Europe, the net factor of safety for wires under the worst conditions of loading is about 2^. When the "flexible" type of tower construction is used, it is customary to allow a somewhat higher factor of safety for the conductors than when the towers are of the so-called rigid type. UEJNAHY FIG. 109. Type of joint for overhead conductors. (0*5' 302 ELECTRIC POWER TRANSMISSION Joints in wires can and should be made of the same strength as the wire itself. A discussion of the various types of joints as used in practice would be out of place in these pages. A very common type is the Mclntyre joint, illustrated in Fig. 109. It is claimed that when the sleeve is long enough to allow of three complete turns, the strength of this joint is equal to that of the cable itself. Under favorable weather conditions it is pos- sible to make very good butt- welded joints on aluminum con- ductors with the aid of a blow lamp. Such joints would probably not be satisfactory on conductors of large cross-section (more than % in. diameter), and they should be made at the insulator where they will not be subjected to the maximum tensile stress. The tie wire usually secures the conductor to the insulator in such a way as to prevent as far as possible the creeping of the con- ductor from one span to another. In some cases, however, it is advisable to allow the conductor to slip, or the tie wire to break, before the tension in the cable is great enough to damage insulator or cross-arm. Soft or semi-hard wire is generally used for ties, and the size should not be less than No. 4 B. & S. copper, or No. 2 B. & S. aluminum. The tie wire should be of the same material as the conductor. Some forms of tie include a serving of the tie wire extending an appreciable distance on each side of the insulator; the object being to afford some protection to the conductor in the event of an arc striking over the insulator. A long sleeve over line wires at point of support is objection- able. It is true that breakages of wires caused by too great rigidity at points of support are of rare occurrence; but where mechanical forms of cable clamps are used, as with the suspen- sion type of insulator, and in many of the larger pin type insu- lators, the method of clamping the conductor to avoid mechanical injury or weakening due to vibration or to swaying of the wires in a wind, should receive careful attention. CHAPTER X TRANSMISSION-LINE SUPPORTS 142. General Considerations. Types of Transmission-line Supports. The supporting poles or structures for overhead electric power transmission lines are of various kinds. Where the ordinary wood telegraph pole or a larger single pole of similar type is not suitable, double poles of the "A" or "H" type, or even braced wooden towers of considerable height and strength, may sometimes be used with advantage. Under certain condi- tions it may be economical to use steel poles of the tubular type, or light masts of latticed steel, even for comparatively short spans; and poles of reinforced concrete have much to recommend them. But for long spans, and the wide spacing of wires necessary with the higher pressures, steel towers, either of the rigid or flexible type, will generally be required. The use of wood poles is therefore limited to comparatively low pressures, and spans of moderate length. It is probable that, in rough country where suitable timber is plentiful, and the cost of transporting steel towers would be high, wooden supports of the "A" or "H" type might be used economically for voltages up to 60,000, even if two three-phase circuits were carried on one set of poles; but for higher voltages the steel tower construction with fairly long spans, would in almost every case, be preferable. The case of a 100,000-volt three-phase line supported on wood poles was mentioned in Chapter I as an example of economic wood-pole construction for a high-voltage transmission. Another example of a wood structure, erected where steel would have been adopted by many engineers as being the only material available, occurs at a river crossing between Astoria and Flavel, Ore. At this point a 1611-foot span is supported on wood-pole towers 135 ft. high, a description of which will be found in the Electrical World oi Sept. 11, 1915. The decision as to the best type of line to adopt is not easily or quickly arrived at. The problem is mainly an economic one, and the decision will depend, not only on the first cost of the 303 304 ELECTRIC POWER TRANSMISSION various types of line construction, but also on the probable life of the line and the cost of maintenance. It is necessary to make up many preliminary estimates of the completed line, and these must obviously include, not only the cost of the various types of supporting structure delivered at points along the line, but also the cost of foundations and erection. Again, even if a suitable kind of wood is readily available in the district to be traversed by the transmission line, it is possible that the cost of seasoning the poles, and treating them with preserva- tive compounds to ensure a reasonably long life, may render the use of steel structures more economical even for comparatively low pressures. The use of latticed steel poles, from 30 to 40 feet high, capable of being shipped and handled in one piece, appears to be gaining favor in districts where ultimate economy over wood poles can be shown to result from the adopcion of these light steel structures. The life of a steel tower line depends somewhat on climatic conditions. In Great Britain the dampness of the climate, together with the impurities in the atmosphere in the neighborhood of manufacturing and populous districts, render light steel structures less durable than in America (except, perhaps, on the Pacific coast, where special precautions are re- quired to guard against rapid corrosion due to the prevalence of fogs and moisture). Not only has the iron- work protected by paint to be repainted on the average every three years, but the spans must usually be short, as the private ownership of valuable property renders the construction of a straight transmission line with long equal spans almost impossible in the United Kingdom. These conditions are all in favor of the employment of selected and well creosoted wood poles, the life of which may be 30 years or more. When making comparisons between wood and steel for trans- mission line supports, it is not only the matter of first cost that has to be considered. Steel structures have the advantage of being invulnerable to prairie and forest fires; moreover, owing to the longer spans rendered possible by the stronger and taller supports, there is less chance of stoppages owing to broken insula- tors, and less leakage loss over the surface of insulators. A fact that is often overlooked is that the size of conductor limits the practical length of span; for instance, with a small conductor such as a No. 4 B. & S., it would not be wise to have spans much above 250 or 300 ft. This suggests what is frequently found to be TRANSMISSION-LINE SUPPORTS 305 the case, namely, that the total cost of a line may be reduced by using a conductor of rather larger section than the electrical cal- culations would indicate as being necessary, because the stronger cable permits of a wider spacing of the supporting towers. The economic length of span on steel tower lines usually lies between 500 and 700 ft., but very much longer spans can be used where the character of the country would render then- use eco- nomical or where rivers have to be crossed. 1 On the transmission system supplying Dunedin City, New Zealand, with electric energy at 35,000 volts, there is a span 1700 ft. long where the line crosses the ravine near the power station. The peculiarity of this span is the great difference in level between the two supports, the upper tower, which is a special steel structure, being 650 ft. above the lower tower. 143. Wood Pole Lines. Kinds of Wood Available. Among the varieties of straight-growing timber used for pole lines on the American continent may be mentioned cedar, chestnut, oak, cypress, juniper, pine, tamarack, fir, redwood, spruce, and locust. In England the wood poles are usually of Baltic pine or red fir from Sweden, Norway, and Russia. The woods used for the cross-arms carrying the insulators include Norway pine, yellow pine, cypress or Douglas fir, oak, chestnut, and locust. Probably the best wood for poles is cedar; but chestnut also makes excellent durable poles. Much depends, however, on the nature of the soil, and, generally speaking, poles cut from native timber will be more durable than poles of otherwise equally good quality grown under different conditions of soil and climate. With the more extended adoption of preservative treatments (to be referred to later), the inferior kinds of timber which under ordinary conditions would decay rapidly, will become of relatively greater value, and with the growing scarcity of the better kinds of timber, it is probable that poles of yellow pine, tamarack, and Douglas fir will be used more extensively in the future. The trees should be felled during the winter months, and after being peeled and trimmed should be allowed to season for a period of at least twelve months. A brief specification covering wood poles for power transmis- sion lines will be found in Appendix II at the end of this book. 1 See Article 140, Chapter IX. 20 306 ELECTRIC POWER TRANSMISSION 144. Typical Wood Pole Lines. For a single three-phase line transmitting power at 20,000 to 22,000 volts single poles having a top measurement of about 8 in. would be suitable. The dis- tance between wires would be about 3 ft.; the arrangement being as shown on the sketch, Fig. 1 10, with pole-top details as in Fig. 111. This shows an arrangement without overhead guard wire, but with some or all poles protected by a grounded light- Fio. 110. Typical wood pole line for pressures up to 22,000 volts three-phase. ning rod. In exposed positions, and at angles, pieces of bent flat iron may be fitted with advantage on the cross-arm near the insulators, as shown by the dotted lines in Fig. 111. These pieces serve the double purpose of hook guard in case of the wire shipping off insulator, and of additional protection against light- ning. A discharge from the line tends to leap across to this grounded metal horn over the surface of the insulator, thus fre- quently preventing the piercing or shattering of insulators. Fig. 112 shows the pole-top arrangement for a 33,000- volt TRANSMISSION-LINE SUPPORTS 307 line of the Central Illinois Company, while the double insulator construction adopted by the same company for corner poles is illustrated in Fig. 113. These two illustrations are reproduced here by kind permission of the Electrical World. A simple "A" frame construction for a duplicate three-phase line operating at 11,000 volts is shown in Fig. 114. Another type of construction for duplicate three-phase line is shown in Fig. 115, where the standard single wood pole is used. This is the arrangement adopted by the Central Illinois Company when it is desired to carry two circuits on the one pole line. Galvanized Iron Lightning Rod Special Pin for Pole-Top Insulator %> Galvanized Steel Cable connecting Lightning Rod to Ground FIG. 111. Pole-top details. 145. Life of Wood Poles Preservative Treatment It is not easy to estimate the probable life of poles, because this will depend not only on the kind of timber, but also on the nature of the soil, climatic conditions, the time of seasoning, whether or not the poles have received treatment with preservative com- pounds, and the nature of such treatment. In England the life of well-seasoned, creosoted poles may be about 35 years in good soil, and from 18 to 20 years in poor 308 ELECTRIC POWER TRANSMISSION FIG. 112. Single-circuit construction for three-phase overhead transmission. FIG. 113. Double-arm corner construction. TRANSMISSION-LINE SUPPORTS 309 soil. On the American continent, where untreated poles have been used in large numbers, the average life is probably about 12 years. The better woods, such as cedar and chestnut, might last on the average 14 to 16 years, while juniper and pine might have to be replaced in 6 to 10 years. On certain lines where un- treated poles of unsuitable timber have been erected in poor soil, or where destructive insects are particularly active, the poles have had to be replaced in less than 4 years. The creosoted poles, as used in England, will usually stand best in moist or clayey imber Tie About 6x9x5 LI FIG. 114. Typical "A" frame construction for duplicate three-phase line. ground; there is a tendency for the creosote to run out and be absorbed into the ground when the soil is loose and sandy, with the result that the poles deteriorate rapidly just below the ground level. Marshy soil is generally bad for wood poles, also ground that is alternately wet and dry. Preservative Treatment of Poles. Many chemical solutions and methods of forcing them into the wood have been tried and used with varying success; but it is generally conceded that 310 ELECTRIC POWER TRANSMISSION treatment with coal-tar creosote oil gives the best protection against decay; and its cost is probably lower than that of any other satisfactory treatment. There are three recognized methods of applying oil : (a) The high-pressure treatment (Bethel system). (b) The open-tank treatment. (c) Brush treatments. In Europe the treatment known as the Riiping process is largely used; it is less costly than the Bethel system. In France This Bayonet for CurYft in line '- *4<---jV -*-- Uf Fio. 115. Double-circuit construction. copper sulphate is used extensively as a preservative (Boucherie process), but the results are not entirely satisfactory. HIGH-PRESSURE TREATMENT This is undoubtedly the best, but it is also the most costly. The poles, after being trimmed and framed, are placed in large TRANSMISSION-LINE SUPPORTS 311 treating cylinders capable of being hermetically closed. If the poles are green or wet they are first subjected, in these cylinders, to a steaming process from three to eight hours, the steam being admitted under a pressure of 12 to 20 Ib. The steam is then blown off, and the treating cylinder is exhausted, the vacuum being maintained for a period of one to two hours. Immediately afterward the creosote is forced in under pressure at a tempera- ture of 140 to 200 F. Seasoned timber is not subjected to the steaming process, but the temperature inside the treating cylinder is raised by means of heating coils to about 150 F. prior to the filling process. The poles will absorb from a minimum of 10 Ib. to a maximum of 15 Ib. of oil per cubic foot. The softer and more porous woods will absorb the most oil; but on the other hand, the benefit such woods derive from the treatment in the matter of increased life is more marked than in the case of the closer grained timber. OPEN-TANK TREATMENT The butts of the poles are placed in the creosote oil, which is preferably heated to a temperature of 200 F. to 220 F. They are maintained in this bath for a period of one to three hours, after which they are placed in cold oil for another period of one to three hours. This process will permit of a complete penetra- tion of the sapwood to a height of about 2 ft. above ground level. When properly carried out it is capable of giving very satisfac- tory results. The open-tank process is specially applicable to the treatment of the more durable kinds of timber, such as cedar and chestnut. BRUSH TREATMENT The oil is applied hot with hard brushes, a second coat being applied after the first has soaked in. The temperature of the oil should be about 200 F. This method of application of the oil is the cheapest and the least effective, but it affords some pro- tection when the wood is well seasoned and dry. There is little advantage to be gained by the external application of preserva- tive compounds to green timber; indeed, the sealing up of the surface of such timber, by enclosing the fermentative juices, may even lead to more rapid decay. The brush treatment cannot be applied to poles which are set in the winter months in cold cli- 312 ELECTRIC POWER TRANSMISSION mates, as the frost would so harden the surface of the poles that there would be no absorption of the preservative liquid. The quality of oil used, whatever the method of application, is a matter of importance. In circular No. 98 of the United States Forest Service, Department of Agriculture, issued May 9, 1907, the concluding statement is to the effect that light oils boil- ing below 400 F. will not remain in the timber; but the heavy oils, containing a high percentage of anthracene oil, will remain almost indefinitely, and afford excellent protection against decay and boring animals. The reader who desires to go further into the important ques- tion of wood pole preservation, is referred to the excellent book by Howard F. Weiss, " The Preservation of Structural Timber" (McGraw-Hill Book Company). 1 It seems hardly necessary to point out that the saving effected by prolonging the life of the poles will usually justify the cost of the special treatment. Given reliable data regarding costs and probable life of treated and untreated poles, the necessary calculations are easily made. As an example; if the cost of a pole after treating by the brush process is $24 and the average life 10 years, while the extra cost of open-tank treatment is $1 with an added 10 years to the life of the pole, it is easy to show that the tank treatment will be the more economical in the long run. INFLAMMABILITY OF TREATED TIMBER Poles and cross-arms treated with creosote oil are less liable to destruction by fire than untreated timber of the same kind. This appears to be due to the fact that the free carbon deposited by the burning oil on the surface of the timber affords some pro- tection from the action of the fire. A committee appointed by the National Electric Light Association of New York City conducted a series of experiments on similar specimens of treated and un- treated short-leaf pine, and proved conclusively that the latter suffered considerably more damage from the effects of fire than the specimens that had been impregnated with creosote oil. Reinforcing Pole Butts. Wood poles usually decay most rapidly at or near the ground surface. It is not always necessary 1 See also "Preservative Treatment of Telephone Poles," by F. L. Rhodes and R. F. Hosford, Trans. A. 1. E. E., vol. xxxiv (part 2), p. 2549, Oct., 1915. TRANSMISSION-LINE SUPPORTS 313 to replace poles which may be otherwise sound, but which have been weakened locally by decay just below the ground surface. They may be reinforced by means of steel rods about > in. diame- ter, pointed at both ends and driven into the pole above and below the damaged portion. Concrete is then filled in around the pole, extending at least 12 in. above the ground level. This and similar methods of prolonging the life of poles have proved satisfactory. The cost may be from $3 to $4 per pole, and the life may be prolonged 5 to 10 years. 146. Insulating Qualities of Wood Poles. One advantage which may be claimed for wood poles is the possibility of work- ing on live wires with little danger to life when the conditions are favorable. In the Black Hills in the mining district of South Dakota, where the climate is dry, it is usual to work on live wires at 24,000 volts, supported on ungrounded poles, without using any insulating devices. In this instance, the separation between wires is unusually large, being 5 ft., which affords additional safety. On the other hand, it may be argued that accidental contact with a wood pole carrying high-tension conductors may be a danger to the public, which is practically absent in the case of grounded steel poles or towers. Tests have been made to determine, if possible, the nature of charges likely to pass to ground through a person touching ungrounded wooden poles of a high-tension transmission. These tests show that it is pos- sible for poles to become dangerously charged, but not probable. A grounded metal ring or wire placed around the pole from 6 to 7 ft. above ground eliminates all possibility of accidents from this cause. 147. Weight of Wood Poles. For the purpose of estimating the costs of transport and handling of poles, the weight may be calculated on the assumption that the pole is of circular section and of uniform taper, such that the diameter D at the bottom is equal to the diameter d at the top plus a quantity tH, of which H is the distance between the two sections considered, and t is a constant depending on the taper and therefore on the kind of wood. Some approximate (average) values of t together with average weight per cubic foot of various kinds of dry timber will be found in the accompanying table, from which the value of t for cedar is seen to be 0.0165 (the height H being understood to be expressed in inches'), and the weight per cubic foot, 35 lb.; 314 ELECTRIC POWER TRANSMISSION CONSTANTS FOR WOOD POLES Kind of wood Wt. per cubic foot, Ib. approx. Natural taper t, average Modulus of rupture Modulus of elasticity M" Juniper American Eastern white cedar Spruce White pine 35 27 26 0.0165 3700 4000 4500 4500 700,000 1,300,000 1,000,000 Red pine Douglas fir 3 34 34 5000 6500 1,400 000 Norway pine 7000 1,400000 Redwood Idaho cedar Chestnut 23 42 0.01 0.016 7000 6000 6000 700,000 900,000 1 Being stress in pounds per square inch at moment of rupture under bending conditions. 2 Inch units. Average figures, which must be considered approximate only. 3 This name is intended to cover yellow fir, red fir, Western fir, Washing- ton fir, Oregon fir, North-west and West-coast fir. The volume of a frustrum of right circular cone is: but Volume = o D = d X 4 + Dd + d 2 ) tH, and the formula becomes: Volume = (3d 2 + t*H* + UtH) By using this formula and putting for H the value 65 X 12 = 780 in., and for d the value 7 in., the weight of a pole of American eastern white cedar measuring 7 in. diameter at top and 65 ft. over-all length works out at 2410 Ib. 148. Strength and Elasticity of Wood Poles. Apart from the dead weight to be supported by the poles of a transmission line which will include not only the fixtures and the conductors themselves, but also the added weight of sleet or ice in climates where ice formation is possible the stresses to be withstood include the resultant pull of the wires in adjoining spans, and the wind pressure on poles and wires. It is customary to disregard the dead weight or column loading, except when the spans are large and the conductors numerous and heavy. A formula for approximate calculation .of loads carried by poles when acting as TRANSMISSION-LINE SUPPORTS 315 struts or columns will be given later. The pull due to the con- ductors on corner poles is usually met by guying these poles, by which means the pull tending to bend the pole is largely converted into an increased vertical downward pressure; but even on straight runs there may be stresses due to unequal lengths of span which would cause a difference in the tensions on each side of the pole. The most important stresses to which the poles are subjected, apart from such accidents as are due to falling trees or the sever- ing of all wires in one span, are those caused by strong winds blowing across the line. The resulting pressure at pole-top due to strong winds acting on long spans of ice-coated wires may be very great, and the poles must be strong enough to resist this. 1 For the purpose of making strength and deflection calculations, the pole may be considered as a truncated cone of circular sec- tion, firmly fixed in the ground at the thick end, with a load near the small end in the form of a single concentrated resultant horizontal pull. The calculation is therefore exactly the same as for a beam fixed at one end and loaded at the other. Such a beam, if it exceeds a certain length depending upon the amount of taper, will not break at the point where the bending moment is greatest (i.e., at the ground level), because, in a beam of circular section and uniform taper, the stresses in the material are not necessarily greatest at this point, as will be shown later. The ordinary telegraph or electric lighting pole usually breaks at a point about 5 ft. above ground unless the butt has been weak- ened by decay. Calculations on strengths and deflections of wood poles cannot be made with the same accuracy as in the case of steel structures; and the constants in the table of Article 147 are aver- ages only for approximate calculations. The factor of safety generally used on the American continent is 6, both for poles and cross-arms. The maximum wind pressure is taken at 30 Ib. per square foot of flat surface, or 18 Ib. per square foot of projected surface of smooth cylinders of not very large diameter. In England the factor of safety for telegraph poles is 8, and for power lines 10. The latter figure would seem to be unnecessarily high: it suggests a want of confidence either in the strength calculations or in the tests and load assumptions on which the calculations are based. 1 See Article 131 in Chapter IX. 316 ELECTRIC POWER TRANSMISSION 149. Calculation of Pole Strengths. The relation between the externally applied load and the stresses in the fibers of the wood is: Bending moment = stress in fibers most remote from neutral axis X section factor, or M B = S X Z. If P is the force in pounds applied at a point distant x in. from the cross-section A (see Fig. 116), then: M B = Px Ib.-in. And if the stress S is expressed in pounds p per square inch, and the section is as- sumed circular: Px - S X ^ JTX >J A. ~7T^~ "~Dia. - d + tx But it is assumed that the diameter at ^ any point x in. below the section of diameter d is d X tx, therefore: 32P X (d + (149) FIG. 116. Wood pole with horizontal load near top. In order to find the position of the cross-section at which the pole is most likely to break that is to say, where the fiber stress is a maximum it is necessary to differentiate the last equa- tion with respect to x, and find the value of x which makes this differential equal to zero. This gives d for the point where the stress S is a maximum. The position of this cross-section is evidently not always at ground level. If this value of x is greater than H, then the maximum fiber stress will be at ground level, and it is calculated by substituting H for x in formula (149). The diameter of the pole at the weakest point is: & - d + tx = 1.5 d TRANSMISSION-LINE SUPPORTS 317 and it is only when the diameter at ground level is greater than one and a half times the diameter where the pull is applied that the pole may be expected to break above ground level. If the stress S, the taper t, and the pole-top diameter d are known, the load P is readily calculated as follows: Bending moment = P X x ird 3 w Resisting moment = S X ^ Sm But x = - an d d w = l.5d therefore Pd TT X TTT = o X 2t 32 D 2tSirX 3.375 X d 3 P " 32 Xd = 0.662 X S X t X d 2 (150) Similarly, if the pull P is known, the pole-top diameter should be : -4 0.662 x S X t EXAMPLE OP STRENGTH CALCULATION Consider a pole of Eastern white cedar designed to sustain a pull of 500 Ib. applied 26 ft. above ground level. The average breaking stress (from table of constants) is 4000 Ib. per square inch, and assuming a factor of safety of 6 the safe working stress is S = 660 Ib. per square in. The other numerical values are: P = 500 Ib. H = 26 ft. <(from table) = 0.0165 By formula (151) : Voi 500 / 0.662 X 660 X 0.0165 = 8.33 in. d w = 1.5 X 8.33 = 12.5 in. The distance below point of application of load of the section where fiber stress is a maximum is: x = - = 252 in. = 21 ft. at 318 ELECTRIC POWER TRANSMISSION Therefore this pole, if subject to a load about six times greater than the maximum working load, may be expected to break 26 - 21 = 5 ft. above ground level. Double pole supports of the type illustrated in Fig. 114 will be twice as strong as each of the component poles in resisting stresses applied in the direction of the line; but they will be able to with- stand about five times as great a load as the single pole when the stresses are in a direction at right angles to the direction of the line. When loaded in this manner up to the breaking point, these double poles of the "A" type usually fail through the buckling of the member in compression due to initial want of straightness. The strength of both the "A" and the "H" type of pole structure can to some extent be increased by judi- cious and rigid bracing. 150. Deflection of Wood Poles. It is now generally recognized that there are advantages in having transmission-line supports with flexible or elastic properties. The ordinary single wood pole is very elastic, and will return very nearly to its original form after having been deflected considerably by abnormal stresses. The figures given for the elastic modulus in the table previously re- ferred to are subject to correction for different qualities and samples of the same timber. It is well to make a few experiments on the actual poles to be used if accuracy in calculated result is desired. The double-pole structures of the " A " or " H " type will have about half the deflection of the single poles in the direction of the line, and, of course, very much less in a direction at right angles to the line. An "A" pole of usual construction with the two poles subtending an angle of 6^ degrees will deflect only about one-fiftieth of the amount of the single-pole deflection under the same transverse loading. The movement is usually dependent upon the amount of slip between the two poles at top, which again depends upon the angle subtended by the poles. 1 If this angle is as much as 10 degrees there will be practically no likelihood of the poles slipping at the top joint; but this large angle is unsightly, and probably makes a less economical struc- ture than the more usual angle of about 6^ degrees. 151. Calculation of Pole Deflections. Assume the pole to be fixed firmly in the ground, and that there is no yielding of founda- 1 Much useful information on the behavior of "A" poles under test is to be found in Mr. C. Wade's paper read before the Institution of Electrical Engineers on May 2, 1907. TRANSMISSION-LINE SUPPORTS 319 tions. The load P being applied in a horizontal direction at the top end, as indicated in Fig. 117, the pole may be considered as a simple cantilever, the deflection of which, if the section were uniform throughout the entire length, would be: PH S 3MI where 5 and H are in inches; 7 is the moment of inertia of the section, and M is Young's modulus (pounds per square inch). For a circular section I = -^- where d is the diameter of the (cylindrical) pole in inches. The formula then becomes: 6 Md< If P is evenly distributed, as would be the case with a uniform wind pressure on the pole surface, regardless of other loads, the deflection would be: "SMI but it is best to consider the wind pressure on pole surface as a single equivalent load concentrated at pole-top and added to the load due to wind pressure on the wires. When estimating the probable value of this equivalent load, it should be remem- bered that the wind pressure is not evenly distributed along the length of the pole, since the wind velocity at .ground level is comparatively small and increases with the height above ground surface. The formula (152) assumes a constant diameter throughout length of pole, and the question therefore arises as to where the measurement of diameter should be made on an actual pole. Mr. S. M. Powell has shown that, on the assumption of a uni- form taper, the quantity d 4 in formula (152) should be replaced by (d g s X di) where d g is the diameter at ground level and di is the diameter where the force P is applied. FIG. 117. Deflection of wood pole. 320 ELECTRIC POWER TRANSMISSION EXAMPLE OP CALCULATION OF POLE-TOP DEFLECTION Using the same figures as in the example of strength calcula- tions : P = 500 Ib. H = 26 X 12 = 312 in. di = 8.33 in. t = 0.0165 d = 8.33 + (0.0165 X 312) = 13.48 in. M = 700,000 then _ 6.78 XPXH 3 ' M (<4 3 X di) 6.78 X 500 X (312) 3 700,000 X (13.48) 3 X 8.33 = 7.2 in. When possible it is well to make tests on a few actual poles; then for similar poles of the same material subject to the same loading: 5cx d7x~Ti 152. Pole Foundations. A permanent deflection of the pole when the stresses are abnormal may occur owing to the yielding of the earth foundation; but this is unusual if the poles are properly set in good ground. The diagram Fig. 118 has been drawn to show the depth to which poles of various heights are usually set. These depths are such as would be adopted on a well-designed pole line, and need not be exceeded except in special cases. In marshy or otherwise unsatisfactory ground, special means must be adopted to provide a reasonably good setting for the pole butts. Loam and gravel, and even sand; or a mixture of these, pro- vides a firm foundation for poles. A pole that is properly set should break before the foundations will yield to any appreciable extent. Even if there should be a movement of the pole butt in the ground with excessive horizontal load at pole-top, this will result in a firmer packing of the earth, which will then be better fitted to resist any further movement. Firm sand, gravel, or loam, will withstand a pressure of about 4 tons per square foot; but only half this resistance should be reckoned on in the case of damp sand, moist loam, or loose gravel. Proper supervision is necessary to ensure that the earth shall TRANSMISSION-LINE SUPPORTS 321 be packed firmly around the pole when refilling the holes. This matter of tightly packing the dirt around the pole butt is referred to in Appendix II. Although no attempt has been made to treat adequately, in this book, of practical details such as the best method of digging holes, it may not be out of place to mention that the use of dynamite for digging post holes appears to have met with success where the method has been carefully studied and intelligently applied. (Refer to the Electrical World, June 8, 1912 and Feb. 7, 1914.) vurc A- Poles set in rock Carve B- Poles set in solid ground on straight Carve C - Poles set in solid ground at corners or in poor soil on straight runs Carve D- Poles set at corners in soft ground 25 30 36 40 45 50 5 Height of Pole above Ground .Level - Feet FIG. 118. Chart giving approximate depth of holes for wood poles. 60 153. Spacing of Poles at Corners Guy Wires. In order to reduce the stresses, not only on the pole itself, but also on the insulator pins and cross-arms, it is usual to shorten up the spans on each side of the corner pole. The reduction in length of span will depend upon the amount by which the direction of the wires departs from the straight run. A rough and ready rule is to reduce the span length 1^ per cent, for each degree of deviation from the straight line. For angles less than 5 degrees, it is not necessary to alter the span. It is not advisable to turn more than 25 degrees on one pole, and whenever the side strain is likely to be excessive, double cross-arms and insulators should be used. By giving proper attention to the matter of guying and to the mechanical con- 322 ELECTRIC POWER TRANSMISSION struction generally, it is not difficult to meet all requirements at points where a change of direction occurs. A safe plan is to assume that a corner pole must carry the full load without breaking if the guy wire or wires should fail to take their proper share of the load: but all corner poles should be propped or guyed for extra safety, and to avoid the unsightly appearance of poles bent under heavy side stresses or set at an angle with the vertical. Sometimes when sharp corners have to be turned, the spans on each side are " dead-ended " on poles with double fixtures. Such poles are head-guyed, and the span adjoining the guyed pole is usually shortened, being not more than three-fifths of the average spacing. For further particulars of common practice in guying poles in special positions, the reader is referred to the sample specification for wooden pole line in Appendix II. w FIG. 119. Diagram of stresses at corner pole. The non-synchronous swaying of wires in a high wind, al- though uncommon, sometimes occurs on wood pole lines, being aggravated by the difference in the natural period of oscillation of poles and wires. This trouble can generally be cured by guy- ing one or more of the poles at the place where the wires have been found to swing non-synchronously. Guy wires should be of galvanized stranded steel cable, the breaking strength of which should preferably not exceed about 34 tons per sq. in. The reason for this limitation of strength is that the high-strength steel is usually too hard to allow of proper handling and finishing off. 154. Load to be Carried by Corner Poles. If P is the total tension in pounds of all the wires on each side of the corner pole, and if 6 is the angle of deviation as indicated in Fig. 119, then the resultant pull in the direction OW at the pole top will be, W (Ib.) = 2P sin (153) The stress in the guy wire is readily calculated when the angle TRANSMISSION LINE SUPPORTS 323 a (Fig. 120) which the wire makes with the vertical is known. If W is the side pull as calculated by formula (153), then W Tension in guy wire = CD Q54) 155. Props or Struts Wood Poles in Compression. Some- times it is difficult or impossible to provide guy wires in certain locations; or impurities in the atmosphere may render the use of props or push braces preferable to guy wires. In such cases it is necessary to know approximately what load a wooden pole will support in compression, that is to say when used or considered as a column. Instead of using the values of unit stress, S, as c D FIG. 120. Diagram of stress in guy wire. given in the Table on p. 314, the ultimate stress which a wood column will stand in compression should be calculated by the empirical formula: Stress in compression (Ib. per sq. in.) = S (1 ^^j (155) where I is the length in inches, and d is the diameter or least thickness at the center of the strut. Example. Calculate the safe load for a prop of Douglas fir, 8 in. diameter and 10 feet long, assuming a safety factor of 6. By formula (155) the breaking stress will be 6500 (l - nn _ J) and the maximum safe load will be ~- X ~ X 64 = 41,000 Ib. 324 ELECTRIC POWER TRANSMISSION 156. Reinforced Concrete Poles. As substitutes for wood poles supporting overhead wires, steel poles of the tubular form and latticed steel masts are used. The full advantage of the galvanized or painted steel structure is best realized in the high towers with extra wide spacing, such as are used for the transmission of electric energy at high pressures. The use of Portland cement for moulded poles of moderate height is by no means new; the experimental stage has long ago been passed, and with the deplorable but no less rapid depletion of our forests and the incomparably longer life of the concrete poles, these will probably be used in increasing numbers during the next few years. There is much to be said in favor of the wood pole when the right kind of timber, properly seasoned and treated, is used; but, apart from the general unsightliness of wood poles in urban districts, their life is uncertain and always comparatively short. In Switzerland the experiment has been tried of covering the ordinary wood pole with concrete mortar about 1 in. thick. The strength, and especially the life, are greatly increased thereby, as the decay which so frequently occurs at ground level will be largely, if not entirely, prevented; but it is doubtful whether the system will, in the long run, prove satisfactory or economical. The ideal material to use for reinforcing concrete is undoubtedly steel or iron. Longitudinal rods or bars of iron can be placed exactly where required to strengthen those parts of the pole sec- tion that will be in tension, and the concrete, filling up the spaces between the reinforcing rods, takes the place of all bracing and stiffening members of the ordinary steel structure in an almost perfect manner. It is probably at this time generally admitted that iron embedded in cement will last almost indefinitely without suffering any deterioration. When excavating for the founda- tions of the new General Post Office in London, England, some old Roman brickwork was discovered in which the hoop-iron bonds were still bright and in perfect condition. The life of a concrete pole is, in fact, almost unlimited, a consideration which should riot be overlooked when estimating the relative costs of different kinds of supporting structures. It requires no painting and practically no attention once it is erected. If any small cracks should at any time develop, they can readily be filled with cement. While referring to the advantages of the cement pole it may be TRANSMISSION-LINE SUPPORTS 325 added that every pole is virtually a lightning rod, an advantage which it shares with the steel pole or tower. On lines where both timber and concrete poles have been used and where many wood poles have been shattered by lightning, the concrete poles have rarely been struck. There is an instance of a concrete pole of the Marseilles (111.) Land & Water Company having been struck, but the only damage done was the chipping out of a small piece at the top of the pole and one at the bottom where the current entered the ground after following down the steel reinforcing bars inside the pole. 157. Weight and Cost of Concrete Poles. The weight of concrete poles is necessarily considerable, and unless the poles are made near the site where they will be erected the cost of transportation will generally be prohibitive. Concrete poles usually measure 6 in. at the top with a base width of 10 in. to 14 in. depending on the height. It is, however, quite permissible to use poles with 5 in. top measurement, in which case the base measurement might be about 9 in. for poles not exceeding 25 to 30 feet overall length. The weight of concrete is about 150 Ib. per cubic foot, and the cost of poles will range between 35c. and 70c. per 100 Ib. Should it be found that conditions of labor, transportation, etc., are such that the cost would be in excess of 70c. per 100 Ib. it is probable that steel or wood supports would prove more econom- ical than reinforced concrete. On the basis of 50c. per 100 Ib. weight of the finished pole, the following figures indicate roughly the approximate cost of concrete poles. The lengths given are the overall lengths, including the portion buried in the ground. The weights are such as might be expected for poles designed with hollow cores. Length, ft. Weight, Ib. Cost 30 1800 $9.00 35 2200 11.00 40 3600 16.00 The great weight of concrete poles is probably the most serious objection to their more general adoption in the place of wood poles, where the latter are not readily obtainable or where their appearance is unsightly. 326 ELECTRIC POWER TRANSMISSION It is probable that the concrete poles of cross-country transmis- sion lines are usually made somewhat heavier than the strength requirements necessitate because, being moulded on site, not always with the best and most convenient appliances, they are made solid throughout or through a large part of their length, whereas a hollow construction would have been adopted had suitable collapsible cores been available. Poles up to 35 ft. in length are usually moulded in a horizontal position, the forms being removed after three or four days. After a period of seasoning lasting from two to three weeks they are erected in the same manner as wood poles. Poles longer than 35 ft. are often moulded in a vertical position. The forms are set up immediately over the hole previously pre- pared for the pole base. They are set truly vertical and tempo- rarily guyed, the reinforcing inside the form being held together and in position by whatever means of tying or bracing may be adopted. Sometimes iron wire is used, but more uniform results are obtained by using specially designed iron distance pieces with the required spacing between them. The concrete is raised to the top of the mould by any suitable and economic means (pref- erably direct from the concrete mixer by an arrangement equiva- lent to the ordinary grain elevator) and is dropped in. By this means the hole in the ground is entirely filled with concrete. No tamping is required, a firm hold being obtained, since the ground immediately surrounding the concrete base has not been disturbed. The best quality of crushed stone and sand should be used, the usual proportions being: cement, one part; sand, two parts; crushed stone, three or four parts, not too large to pass through a %-in. screen. The mixture used for the poles on the Pennsyl- vania Railroad is 1.5:2:4. When gravel is used the mixture may be one part of Portland cement to five parts of gravel, pro- 'vided that the latter is graded, including sand, and with the largest pieces of a size to pass through a %-in. screen. The cost of concrete poles, when the long life and other ad- vantages are taken into account, does not compare unfavorably with that of other types; but it must not be overlooked that the cost of materials and labor required to manufacture the poles does not represent the cost of the finished pole erected in position. Much valuable information on the costs of manufacture and hand- ling of concrete poles, together with practical details relating to TRANSMISSION-LINE SUPPORTS 327 methods of manufacture, will be found in Mr. R. A. Lundquist's book on Transmission Line Construction. 1 The reader is also referred to an article by Mr. J. G. Jackson who describes in the Electrical World of Jan. 17, 1914, how the concrete poles used on the Toronto Hydro-electric system were manufactured. The article by Mr. R. D. Coombs, in the Electrical World of Feb. 6, 1915, and a more recent article entitled "Concrete poles carry 22,000-volt power line," in the Electrical World of Feb. 9, 1918 (Vol. 71, p. 296) should also be consulted by those desiring fur- ther information on concrete-pole transmission lines. As an example of a concrete-pole line, the transmission line of the Northern Illinois Light and Traction Company, of Mar- seilles, 111., may be mentioned. This company transmits three- phase energy at from 30,000 volts to 33,000 volts. Most of the poles used are about 30 ft. high, spaced from 125 ft. to 132 ft. apart. The section is square, with 6-in. sides at the top of the pole and 9 in. at the base. The reinforcing consists of six ^-in.- square steel bars through the entire length of the pole. Many of the concrete poles on this line have now been in position over nine years, and they have given entire satisfaction. 158. Strength and Stiffness of Concrete Poles. When design- ing a concrete pole to withstand a definite maximum horizontal load applied near the top, the pole is treated as a beam fixed at one end and loaded at the other. The calculations are very simple if certain assumptions are made, these being as follows: (1) Every plane section remains a plane section after bending. (2) The tension is taken by the reinforcing rods. (3) The concrete adheres perfectly to the steel rods. (4) The modulus of elasticity of concrete is constant within the usual limits of stress. The ultimate crushing stress of the concrete may be taken at from 2000 to 2600 Ib. per square inch. The reinforcing bars should be covered with concrete to a depth of not less than 1 in. The effect of keeping the reinforcing bars under tension while the concrete is poured in the mould and until it has hardened suffi- ciently to support the strain itself has been tried and found to im- prove the performance of the poles, but it is doubtful whether the extra apparatus and labor required are justifiable on economic grounds. When subjected to excessive load a concrete pole will generally yield by the crushing of the material in the base near 1 McGraw-Hill Book Co. 328 ELECTRIC POWER TRANSMISSION ground level; but, unless it is pulled out of its foundations, it will not fall to the ground. The comparative rigidity of concrete poles cannot be said to be a point in their favor, as the flexibility and elasticity of wood poles and some forms of steel structures are features of undoubted advantage under certain conditions. On the other hand, the degree of deflection of concrete poles before breaking is remarkable. The elastic limit is variable, and no exact figure can be given for the elastic modulus of cement concrete; it may be as low as 1,000,000 but for a 1:2:4 mixture 2,000,000 may be taken as a good average figure for approximate calculations. Some tests made on 30-ft. con- crete poles gave deflections of from 3 in. to 4 in. at a point near the top of pole, when subjected to a test load equal to about double the maximum working load. 1 Another series of tests made recently in England on some 44-ft. poles of hollow section, 17 in. square at the base and 8 in. square at the top (inside dimensions 13 in. and 4 in. respectively), with loads applied 38.5 ft. above ground level, gave a deflection of 66 in. under a horizontal load of 10,500 lb., and the permanent set on removal of load was 21 in. The pole did not fail completely until the deflection was 78 in. The illustration Fig. 121 shows a typical concrete pole of hollow section suitable for carrying six transmission wires on two wooden cross-arms. The pole is 35 ft. long over all, about 6 ft. being buried in the ground. With a top measurement of 7 in. square and a taper to give an increase of 1 in. width for every 5 ft. of length, the size at the bottom will be 14 in. square. The 1 These poles were probably of large cross-section. Some tests made on poles measuring 10 in. square at the base and 32 ft. high gave a deflection of just over 2 ft. with a horizontal load of 2000 lb. applied near the top. Section A-A Fia. 121. Concrete pole of hollow section. TRANSMISSION-LINE SUPPORTS 329 drawing shows a section through the hollow pole taken at a point about 4 ft. above the ground level. Iron spacing pieces, as here shown, or their equivalent, must be placed at intervals to hold the longitudinal steel reinforcing bars in the proper position. The number of rods will vary with the distance below the point of application of the load. The bending moment to be resisted at every point being known and the taper of the pole decided upon, the amount of reinforcing required at any given section is easily calculated. The weight of a pole as illustrated would be about 2700 Ib. without fixtures. The reinforcing rods and spacing rings would account for approximately one-seventh of the total weight. A factor of safety of four is generally employed in strength calculations of reinforced concrete poles. In some cases the calculations have been based on a safety factor of 5; but there appears to be no justification for using so large a factor. 159. Steel Poles and Towers Introductory Remarks. It can- not be said that there is at the present tune one standard type of steel structure for supporting the conductors of overhead trans- mission lines; neither is it likely that one particular design will ever be found suitable for all countries, climates and voltages. Any kind of supporting structure which will economically fulfil the necessary requirements will answer the purpose of the trans- mission line engineer, who merely requires a durable mechanical structure to carry a variable number of insulators at a height above ground, and with a spacing between them, depending upon the voltage of transmission and the length of span. As a substitute for wood poles, steel tubes have been used, either in one piece, or built up of several sections of different sizes in order to economize material and give a large diameter at the bottom where the bending moment is greatest, and a small diameter at the top where the bending moment is negligible. Steel poles of considerable height, suitable for longer spans, may be built up of three or four vertical tubes of comparatively small diameter joined and braced together at suitable intervals to give stiffness to the structure. It is doubtful whether, in the long run, such composite tubular structures will hold their own against the small-base latticed steel masts built up of standard sections of rolled steel, as used extensively on the continent of Europe, and, to a relatively smaller extent, in America. The term "tower" is applied mainly to the light steel structures in which the spac- 330 ELECTRIC POWER TRANSMISSION ing between the main upright members, at ground level, is large compared with the height of the structure; the usual proportion which will generally be found to be the most economical in material being 1 to 4; that is to say, if the base is square, the side of this square will be about one-quarter of the distance from the point of measurement to the top of the tower. If the towers are large, the footings are usually separate pieces which are cor- rectly set in the ground by means of a templet, and to which the legs of the tower proper are afterward bolted. A good example of large steel towers is to be found in the 100,000-volt trans- mission line of the Great Western Power Co. of California. Two three-phase circuits are carried on these towers, the vertical spacing between the cross-arms being 10 ft. There are three cross-arms, each carrying two conductors one at each end. The horizontal spacing between wires is 17 ft. on the two upper cross-arms and 18 ft. on the lower cross-arm, which is 51 ft. above ground level. No conductor is closer than 6 ft. 5 in. to the steel structures, this being the minimum clearance in the horizontal direction. The average distance between towers is 750 ft., and they are joined at the top by a grounded guard wire 5 ft. above the bottom of the highest cross-arm. The base of the tower measures 17 ft. square, the parts under ground being separate pieces of steel, buried to a depth of 6 ft. to which the tower proper is bolted after being assembled and erected on site. Although the larger towers are nearly all built of the square type as used for windmills, there is a notable exception in the case of the 140,000- volt line in Michigan, where the towers are of a special three-legged type, built up entirely of angle sections. Fig. 122 shows a typical form of small-base latticed steel mast on the transmission lines of the Iowa Railway and Light Com- pany, Cedar Rapids, Iowa; while Fig. 123 is a good example of square base tower carrying two three-phase lines. These illustrations are reproduced from photographs kindly supplied by the Ohio Brass Co. of Mansfield, Ohio. The large towers of Fig. 123 were designed and constructed by the American Bridge Co. of Pittsburg for the American Gas and Electric Company's 130,000-volt transmission between Wheeling, W. Va. and Canton, Ohio. The six conductors are each of 200,000 circular mil cross-section, and the two grounded guard wires are of the same TRANSMISSION-LINE SUPPORTS 331 size. The line is 55 miles long, and the average length of span is 580 feet. The economical span for the square latticed poles, of the type shown in Fig. 122, is probably something less than 450 feet; but for comparatively light lines, this form of structure with spans of 400 to 430 feet is very satisfactory. 160. Flexible Towers. Although calculations of stresses in transmission lines are usually based on the assumption that the ends of each span are firmly secured to rigid supports; this con- dition is rarely fulfilled in practice; there is some "give" about the poles or towers, especially when the line is not absolutely straight, and the insulator pins will bend slightly and relieve the stress when this tends to reach the point at which the elastic elongation of the wires will be exceeded. Then, again, the wires will usually slip in the ties at the insulators, even if these ties are not specially designed to yield or break before damage is done to the insulators or supporting structures. The use of the suspen- sion type of insulator, which is now becoming customary for the higher voltages, adds considerably to the flexibility of the line. In regard to the towers themselves, all steel structures for dead-ending lines or sections of lines are necessarily rigid, and the usual light windmill type of tower with wide base is also without any appreciable flexibility. The latticed steel masts, as used more generally in Europe than in America, are slightly more flexible, and the elastic properties of the ordinary wood pole are well known. The deflection of a wood pole may be considerable, and yet the pole will resume its normal shape when the extra stress is removed. There is much to be said in favor of so-called flexible steel structures; that is to say, of steel sup- ports designed to have flexibility in the direction of the line, without great strength to resist stresses in this direction; but with the requisite strength in a direction normal to the line, to resist the side stresses due to wind pressures on the wires and the supports themselves. Such a design of support has the important advantage of being cheaper than the rigid tower construction, in addition to which it gives flexibility where this is advantageous, with the necessary strength and stiffness where required. The economy is not only in the cost of the tower itself but in the greater ease of transport over rough country, the preparation of the ground, and erection. 332 ELECTRIC POWf/R TRANSMISSION The advantages of flexibility in the direction of the line are considerable. Probably the most severe stresses which a trans- mission line should be capable of withstanding are those due to the breakages of wires. Such breakages may be caused by ab- normal wind pressures, by trees falling across the line, or by a burn-out due to any cause. Suddenly applied stresses such as are caused by the breaking of some or all of the wires in one span are best met by being absorbed gradually into a flexible system. The supports on each side of the wrecked span will bend toward the adjoining spans because the combined pull of all the wires in the adjoining spans is greater than the pull of the remaining wires, if any, in the wrecked span. This movement of the pole top results in a reduction of tension in the wires of the adjoin- ing span owing to the increased sag of these wires; there will be an appreciable deflection of the second and third poles beyond the break, but the amount of these successive deflections will de- crease at a very rapid rate and will rarely be noticeable beyond the fourth or fifth pole. It is obvious that, as the remaining wires in the faulty span tighten up, the stress increases; but the combined pull of these wires on the pole top is smaller than it was before the accident, since it is assisted by the pull of the deflected poles, and these joint forces are balanced by the combined pull of all the wires in the adjoining sound span, which pull, as previously mentioned, is smaller than it was under normal conditions. The greater the flexibility of the supports in the direction of the line, the smaller will be the extra load which any one support will be called upon to withstand; on the other hand, it is usual to provide anchoring towers of rigid design about every mile on straight runs, and also at angles, in addition to which every fifth or sixth flexible tower may be head-guyed in both directions. In the writer's opinion, too much stress is usually laid on the necessity for providing rigid strain towers at frequent in- tervals to prevent the effect of a break in the wires, or the failure of a single support travelling along the line and causing in- jury to an indefinite number of consecutive spans. The semi- flexible structures referred to are not designed, or should not be designed, without very careful consideration of the conditions they have to fulfil; and there appear to be no scientific reasons, and no records of injury to actual lines, which would justify the assumption that transmission lines of this type are liable to FIG. 123. Typical steel tower transmission line. (Facing page 332) FIG. 124. Flexible steel tower line. TRANSMISSION-LINE SUPPORTS 333 be wrecked in the same wholesale or cumulative manner as a row of card houses. The strain towers are undoubtedly helpful at the time when the wires are strung; but it is possible that they are used at more frequent intervals than the economies of sound engineering require. In level country, a modified clamp in the form of a sleeve with flared ends may be used in conjunction with the lighter (and cheaper) flexible type of supporting structures; and a com- promise between the loose sleeve and the rigid clamp or tie can be used on all lines with flexible supports. Clamps of this type are designed to allow the wire to slip before the combined pull of all the wires exceeds the load that will permanently de- form the supporting structure; and although it is almost impos- sible to ensure that such devices will remain for any length of time in the same condition as when they are installed, yet they will generally afford a reasonable degree of protection in the event of the simultaneous breaking of all the wires in one span. It is not unusual to carry a galvanized Siemens-Martin steel strand cable above the high-tension conductors on the tops of the steel structures. This has the double advantage of securely, but not rigidly, tying together the supports, and of providing considerable protection against the effects of light- ning. The disadvantages are increased cost and possible but not probable danger of the gounded wire falling on to the con- ductors and causing interruption of supply. The dead-end towers should be capable of withstanding the combined pull of all the wires on one side only, when these are loaded to the expected maximum limit, without the foundations yielding or the structure being stressed beyond the elastic limit. The flexible supports must withstand, with a reasonable factor of safety, the dead weight of conductors, etc., and the expected maxi- mum side pressures; but in the direction of the line their strength must necessarily be small, otherwise the condition of flexibility cannot be satisfied. It is easy to design braced A-frame or H-frame steel structures of sufficient strength to withstand the dead load and lateral pressure and yet have great flexibility, with correspondingly reduced strength, in the direction of the line. Great care must be used in designing a line of this type so that strength and dura- bility shall not be sacrificed to lightness and flexibility without very carefully considering the problem in all its aspects. 334 ELECTRIC POWER TRANSMISSION The assumptions made for the purpose of simplifying strength calculations are not always permissible. For instance the effect of a twist in these flexible structures is sometimes overlooked; but when there is inequality of tension in the wires on the two sides of the structure, the fact that the section passing through the two main upright members is no longer a plane at right angles to the direction of the line accounts for the lessened strength of such a flexible design to resist loads due to high winds blowing across the line. It is not safe to adopt the flexible type of transmission line support without expert advice and adequate engineering supervision. As an approximate indication of pres- ent-day practice in arriving at the load in the direction of the line for which flexible structures should be designed, it may be stated that a load of from one-twentieth to one-tenth of the total load for which the rigid-strain towers are designed should not stress the intermediate flexible structures beyond the elastic limit. It is well to bear in mind that at the moment of rupture of one or more wires on a "flexible" transmission line the resulting stresses in the structures and remaining wires will be in the nature of waves or surges until the new condition of equilibrium is attained, and the maximum stresses immediately following a rupture will generally exceed the final value. The mathematics required for the exact determination of stresses and deflections in a transmission line consisting of a series of flexible poles is of a very high order, even when many assumptions are made which practical conditions may not justify; but the limiting steady values of these stresses and deflections can be calculated in the manner described in Article 170 at the end of this chapter, and as the range between these limits will usually be very small, the probable maximum stresses under given conditions can be estimated with a reasonable degree of accuracy. The illustration Fig. 124 kindly supplied by Messrs. Archbold Brady and Co., shows a common form of "flexible" high voltage transmission line following a railway. 161. Steel Poles for Small Short-distance Transmission Schemes. As a substitute for wood poles, light steel structures that can be shipped and erected in one piece appear to be gain- ing favor. Small amounts of energy at comparatively low voltages can be transmitted over distances of 20 to 30 miles by overhead wires supported on steel poles at a cost which need be no higher, and is sometimes even lower, than if the less durable FIG. 125. Bates one- piece expanded steel transmission pole. FIG. 126. Type of steel pole manufactured by the Carbo Cor- poration. (Facina page 334) TRANSMISSION-LINE SUPPORTS 335 and less sturdy wood-pole construction is adopted. One type of steel pole for small lines is the Bates One Piece Expanded Steel Truss of which Fig. 125 is an example. These poles are manufactured by the Bates Expanded Steel Truss Co. of Chi- cago, with the idea that wooden poles are gradually giving out and that this is a practical substitute, the pole being made in one piece without bolted or riveted lattice work. A pole of this type is easily and economically painted. Another make of light- weight steel pole is shown in Fig. 126. This is manufactured by the Carbo Corporation of Chicago, which makes a specialty of steel poles for the economical construction of moderate voltage small-power transmission lines. Steel poles of the type under discussion would range from 25 ft. to 35 ft. in height and would be spaced from 250 to 300 feet apart. 162. Loads to be Resisted by Towers. The maximum load which a tower must be designed to withstand will depend upon the number and size of wires to be carried and the estimated ice coating and wind velocity. Apart from the wind pressure on the structure itself, the loading in a direction transverse to the line will be equal to the resultant wind pressure on all the wires (which may or may not be ice coated, depending on the climate) ; the effective length of each wire being the distance between supports. In the direction of the line, the forces are normally very nearly balanced, but in the event of one or more wires breaking, the unbalanced load may be considerable, and it is well to design the towers, if possible, to withstand the stresses imposed upon them if two-thirds of all the conductors in one span are severed. It must not be overlooked that if the wires break in one span only, the cross-arm, if pin type insulators are used, will be subjected to a twisting moment; and if the break in the wires is at one end only of the cross-arm, the whole tower is subjected to torsional strain. The vertical or dead loads consist of the weight of the tower itself and the wires of one span, with possible increase in weight due to sleet or ice. The cross-arms must be of ample strength to take all vertical loads including weight of insulators, with a mar- gin to cover the extra weight of men working on the tower. The approximate weight of insulators is given in the following table: 336 ELECTRIC POWER TRANSMISSION Working line voltage Weight of insulator, Ib. 22,000 44,000 pin type... ..: (j* 66,000 40 88,000 suspension type 55 110,000 ' 70 140,000 I 90 If a weight is attached to the lowest unit in a string of suspen- sion insulators with the object of limiting the angle of deflection from the vertical when a high wind is blowing across the line, this must be taken into account when determining the loads to be resisted by the supporting structure. Particulars regarding wind pressures were discussed in Article 131 of the preceding Chapter. The wind velocity rarely exceeds 80 miles per hour either on the American continent or in England. Tornadoes and cyclones are not considered, because attempts to design overhead lines strong enough to withstand them, would hardly be justified. In regions where sleet de- posits are to be expected it appears to the writer unreasonable to base calculations on a heavier loading than that described as Class B on p. 271 of Article 131. That is to say, an ice coating 0.5 in. thick is very rarely exceeded, and when the conditions are such as to permit this formation, a wind velocity exceeding 60 miles per hour (corresponding to 8 Ib. per foot pressure) is not likely to occur. In regions where strong winds may be expected, but where sleet deposits do not occur, a maximum wind velocity of 76 miles per hour seems a reasonable assumption. This corresponds to a pressure of 21 Ib. per sq. foot of flat surfaces on towers, and 14 Ib. per sq. foot of projected surface of wires and cylin- drical poles. The total transverse load is dependent upon the length of span, which must be determined with due regard to economic considerations. 163. Design of Steel Towers. Although details of design and the proportioning of parts are matters best left to the manufac- turer, the general type of supporting structure to be used under given conditions should receive careful attention. The most economical design of tower to withstand the probable loads TRANSMISSION-LINE SUPPORTS 337 that it will be subject to, and to satisfy local conditions, includ- ing such considerations as transport and erection facilities, is a problem deserving close attention on the part of the engineer responsible for the design of the transmission line. A study of the probable loads to be resisted under the worst weather con- ditions will enable the designing engineer to specify certain test loads which will ensure that the finished structure will be strong enough to fulfil the practical requirements. The proper value of these test loads and their distribution or point of application should be determined only after mature consideration. The cost of a tower apart from the height, which is a function of the length of span is determined largely by the specifications of test loads. A specification calling for tests that are unneces- sarily severe, is just as true an indication of incompetence on the part of the designing engineer as a specification giving test conditions that will result in a tower too weak for the actual requirements. The calculation of stresses in the various members of so simple a structure as a transmission line tower is not a difficult matter, especially if graphical or semi-graphical methods are adopted. If the designing engineer will make sketches of two or three al- ternative designs likely to fulfil the required conditions, he should be able quickly to calculate the approximate value of the stresses in the principal members, and so obtain a rough idea of the rela- tive weights and costs of alternative designs. The danger of leaving the problem entirely in the hands of the manufacturer is that the latter is always tempted to put forward a design of which he has perhaps made a specialty, and which may have given entire satisfaction in practice without necessarily being the best type of structure for the purpose, or being entirely suitable for use under different conditions. 164. Stresses in Compression Members of Tower Structures. The failure of steel towers under excessive loads is almost invariably due to the buckling of the main leg angles in compres- sion. The designer should therefore pay special attention to the proportioning of compression members in the structure. With- out going into a discussion of the many empirical formulas used for determining the loads that struts or columns can withstand, it may be said that, for tower designs, the "straight line" for- mula, as suggested by Burr, is quite satisfactory provided the ratio I -T- r lies between 40 and 200; this last figure corresponds 22 338 ELECTRIC POWER TRANSMISSION to a length of compression member not exceeding about twenty times the width of flange. This formula is Scomp. = K - k(l/r) (156) where K and k are constants, I is the length, in inches, of unsupported portion of com- pression member, r is the least radius of gyration, in inches, ment of inertia area of section CO mp. is the unit stress (Ib. per sq. in.) in the column. The ultimate stress which will cause compression members of steel towers to collapse is approximately expressed by the formula, Scomp. = 35,000 - 120 l - (157) Assuming a factor of safety of 2^, we may write: Safe working S comp . = 14,000 - 48 l - (158) which may be used in the design of steel towers. A similar formula, which is in common use for calculating safe loads in compression members of steel structures is: Scomp. = 16,000 - 70 l - (159) This formula (159) is a safer one than (158) to use when the ratio l/r is large; but in any case it is recommended that l/r shall not exceed 120 for main members and 150 for lateral or secondary members. The fact that, for a given cross-sectional area, the shape of the section is an important factor in determin- ing the stiffness and ultimate strength of the members in com- pression, suggests that, where lightness and economy of material are of great importance, a section of structural steel having a large moment of inertia per square inch of cross-section should be chosen. The standard sections of rolled angles or tees are some- times replaced by steel tubes. As an example of the relative economy of the tubular form and other forms of section, when used as comparatively long struts, a TRANSMISSION-LINE SUPPORTS 339 FIG. 127. Steel tower with members of tubular section. 340 ELECTRIC POWER TRANSMISSION steel tube 7 in. internal diameter, % in. thick, weighing 10 Ib. per foot, will be as efficient in resisting compression as a steel angle 7^ in. by 7^ in. by J- in. thick, weighing 25 Ib. per foot, or as an I beam 8 in. by 6 in. by ^ in. thick, weighing 35 Ib. per foot. So large a tube would not be required except in very high towers: a tube from 4 to 5 in. diameter would gen- erally be large enough for the main members of a transmission line tower up to 100 ft. high. The illustration, Fig. 127, is from a drawing kindly supplied by Messrs. Stewarts and Lloyds, Limited, of Glasgow, Scotland; it represents a tower 146 feet high as supplied for a power transmission line in the south of England. 165. Outline of Usual Procedure for Calculating Stresses in Tower Members. The illustration, Fig. 128, which is reproduced by kind permission of the Shawinigan Water and Power Co., and the Canadian Bridge Co., Limited, shows a typical square- base galvanized steel tower as used on the Three Rivers line of the Shawinigan Water and Power Co. of Montreal. These towers are designed to carry six aluminum conductors of nineteen strand 200,000 circular mil cable, each being supported by seven suspension disks of the Ohio Brass Co.'s standard type. In addition to the conductors, there are two ground wires of % in. stranded Siemens-Martin steel cable attached to the points (1) at each end of the upper cross-arm. The line is built for 100,000 volts. The method of procedure in calculating stresses is to make a sketch showing the points of application, and the vertical and horizontal components, of the outer forces. Then indicate by arrows the assumed horizontal and vertical components of the reactions, using the suffixes R and L to indicate the direction or assumed direction of the horizontal components. Since the whole structure is in equilibrium under the influence of the various loads and reactions, it is merely necessary to see that the three following conditions are satisfied at any point considered: (a) The sum of all vertical force components = zero. (b) The sum of all horizontal force components = zero. (c) The sum of all moments about any point = zero. When taking moments in any particular plane, all those in a clockwise direction would be considered positive and those in a counter-clockwise direction negative. All joints are considered as frictionless pivots, which assumption is, of course, not strictly TRANSMISSION-LINE SUPPORTS 341 correct, especially in the case of riveted joints. It is usually an easy matter to choose a section through the structure in such a position that the stresses in a given bar can readily be calculated by applying one or more of the three equations of equilibrium. Top Vie Groundwire connection at 2 points-] Aluminum conductor connection at 6 polnts-2,3,&4 Section C- C Towers are designed for the following Loadings I Breast pull of 12000* horizontal, normal or par- allel to line, applied equally at four points 7 and. 10. II 2COO* parallel to line at each of two points 2,3, and 4, on one side of tower. III 3000 parallel to line at one point 1. IV 1100* vertically at each of three points 1-2-3 and 4 V 1100* " eight 12-3 and 4 I and V are simultaneous . Note: All bolts X with sq. nut and std, * 10 gave. B. & 8, washer. Holes punched "/" for K" bolts. All members galvanized. FIG. 128. Steel tower with members of angle section. The sketch, Fig. 129, will serve to illustrate the method usually followed in calculating the stresses in the main members of a tower structure such as the one shown in Fig. 128. The loading considered is that corresponding to the condition of test loads I and V applied simultaneously. 342 ELECTRIC POWER TRANSMISSION The point at which the horizontal breast pull of 12,000 Ib. is applied corresponds approximately to the point 65 ft. above ground level where the corner legs would meet if produced beyond the points (13). The weight of the tower (which it is supposed has not yet been designed in detail) is taken at 4000 Ib., and this, together with the test load V, gives a resultant vertical loading of 12,800 Ib. applied somewhere on the center line of the tower. Consider a section such as XY which cuts only three members, namely, the leg A at ground level, the leg B just above joint 0', and the diagonal brace C. Select a point where the mem- bers A and C meet, and consider the moments, in the plane of the paper, which are produced about this point by the external forces and the reactions in the members severed by the imaginary section XY. It is obvious that the stresses in A and in C have no effect on the tendency of the part of the struc- ture above the section line to rotate on the point 0, and the whole of the externally applied turning moment must be resisted by the stress in the member B. Therefore FK, ik-Sketch for calculation ( 12 > 80 X 5-75) + (12,000 X of stresses in tower members. 47.5) (x X 11.5) = from which it is found that x = 56,000 Ib. Since there are two members B taking the whole crushing stress, the total load tending to crush the one member B is 28,000 Ib. The length of the unsupported portion of this member is 5.5 ft. or 66 in. The cross-section of 4 in. X 4 in. X J4 in. angle is 1.93 sq. in.; and the least radius of gyration, r = 0.79. The test load should not strain the tower beyond the elastic limit. Using the formula (157), the ultimate stress is, ,. 35,000 - 120 (^ : 25,000 TRANSMISSION-LINE SUPPORTS 343 This corner member is therefore capable of supporting, just before collapse, a compressive load of 1.93 X 25,000 = 48,300 Ib. It should be of ample strength to resist the test load of 28,000 Ib. without permanent deformation. Turning now to the uplifting force acting in the member A and tending to pull up the foundation, the center from which the moments are calculated is shifted to the point 0' where the mem- bers C and B meet. The equation of moments is now, (12,000 X 65) - (12,800 X 8) - (x X 16) = whence x = 42,300 and the tension in one corner angle A is 21,150 Ib. The above example briefly describes what is known as the method of moments. It has been assumed that the tower side under consideration lies in the same plane as the external forces: but the error introduced is practically negligible. It is an easy matter, if desired, to make the necessary correction. When calculating the stresses in a diagonal member such as C of Fig. 129, the moments would be taken about the point 0", which is the junction of the members A and B\ but in that case the actual loads on cross-arms and the wind pressure on the side of the tower would have to be taken into account and sub- stituted for the concentrated test load of 12,000 Ib. at the point 0" which does not produce any stress in the brace C so long as the corner angle A remains truly straight and exerts no lateral pres- sure at the point O. The method of moments can usually be applied for all sections of a tower structure if the imaginary dividing planes are properly placed. The counter members or ties that are not in tension under the conditions of loading con- sidered are usually assumed to be non-existent, i.e., to serve no useful purpose as compression members. When computing the stresses in the flexible "A"-frame steel structures it is assumed that the structure remains always normal to the line in a vertical plane; but unbalanced forces in the conductors will actually deflect the frame from this position and so reduce its possible resistance to transverse loads. It is practically impossible to calculate the strength of the distorted frame, and although flexibility in the direction of the line is usually a desirable feature of this type of structure, it is very important to design the so-called flexible steel towers so that they will not be deflected unduly by such torsional loads as they 344 ELECTRIC POWER TRANSMISSION may be subjected to at times when strong winds are blowing across the line. 166. Stiffness of Steel Towers. Deflection Under Load. The deflection of the top of a transmission-line tower of the ordinary light "windmill" type with wide square base, when bolted to rigid foundations and subjected to a horizontal load such as to stress the material to nearly the elastic limit, might be from 2 to 5 in. With regard to the two-legged or "flexible" type of tower, if this is of uniform cross-section, it may be treated as a beam fixed at one end and free at the other end. If the resultant pull can be considered as a single concentrated load of P Ib. ap- plied in a horizontal direction, at a point H inches above ground level, the deflection, in inches, will be, s (160) where M is the elastic modulus for steel (about 29,000,000; being the ratio of the stress in pounds per square inch to the extension per unit length), and / is the moment of inertia of the horizontal section of the structure. 167. Tower Foundations. The upward pull of the tower legs, which was found in the above example to amount to 21,150 Ib., has to be resisted by the foundation. A factor of safety of 2J to 3 should be allowed. The weight of concrete may be taken at 140 Ib. per cubic foot, and of good earth at 100 Ib., the volume of the earth to be lifted being calculated at the angle of repose, which may be about 30 or 33 degrees with the vertical, as in- dicated in Fig. 130. If the footing of a tower is in gravel, or a mixture of sand and loam tightly packed, there is actually a far greater resistance to the pulling up of the footings than that which is offered by the mere weight of the footings with prism of earth as calculated in the usual way. When concrete has to be used, it is generally cheaper to rein- force it with steel of an inverted T form, as this makes a lighter construction than a solid block of concrete, and an equally good hold is obtained owing to the increased weight of the packed earth which has to be lifted. At the same time it must not be forgotten that the digging of a large hole 5 to 8 ft. deep is con- siderably more costly than the digging of a hole about 2 ft. square, and this extra cost in erection must be taken account of in designing the footings. In marshy or loose soil, or where the TRANSMISSION-LINE SUPPORTS 345 right of way is liable to be flooded, special attention should be paid to the design of durable foundations. Concrete footings with or without piles, or rock-filled crib work may be necessary; it is a matter requiring sound judgment and, preferably, previous experience on the part of the engineer in charge of construction. Crumbling hillsides are best avoided; it is extremely difficult to guard against damage by land slides or even snow slides when towers are erected on the steep slopes of hills. The use of concrete adds considerably to the cost of founda- tions and it should be avoided if possible; on the other hand, it is Natural slope of \ earth showing \ cone shaped mass \ resisting upward pull on tower footing. U_._ 2r _.-^-Bedding of large > flat stones FIG. 130. Foundation for steel tower anchor stub. not easy to design foundations to resist a given uplift without an exact knowledge of the soil conditions at the site of the tower. For the greatest economy of foundation, it is necessary that the designer obtain reliable information on this point. Assuming an average angle of slope of 30 degrees, as indicated in Fig. 130, and a weight of soil of 100 Ib. per cubic foot, the depth of, foundation may be calculated as follows. Let h = depth of footing below ground level, in feet. r = equivalent radius of footing area, in feet. R = radius at ground level of conical section of earth to be lifted. (Feet.) 6 = angle of natural slope of earth. The volume of frustrum of cone to be lifted is, V = h(r* + fl 2 + rR) (161) 346 ELECTRIC POWER TRANSMISSION or, if r + h tan be put in the place of R, V = I h(3r 2 + h 2 tan 2 6 + 3r/i tan 0) (162) If e = 30 degrees, tan 6 = 0.5774 and (approximately), V =TT h(r 2 + 0.1U 2 + 0.58r/i) (163) If r = 1 ft., and h = 7 ft.; the volume of earth to be lifted, by formula (163) is then, V = 230, which gives, W = 23,000 Ib. As previously mentioned, if the soil is firm, this method of calcu- lation usually gives results well below actual values of pull re- quired to uplift the footing. Under the conditions upon which this example has been based, it is probable that the footing would not move with a pull appreciably smaller than 30,000 Ib.; there would then be a packing of the soil immediately above the footing, and a final pull of about 40,000 Ib. might be necessary to uproot the stub and footing. If the footings are imbedded in concrete, and separate ground- ing rods are not provided, it is well to let the iron-work project through the bottom of the concrete block, to ensure that the tower is properly grounded. When concrete is not used, the design of the anchors is a matter that should receive very careful consideration. If towers are to be subjected to load tests, these tests should, if possible, be conducted on a tower set on its own anchors, as used in the field, because the strength is to an appreciable extent dependent upon the method of attachment of the tower's legs to the anchor stubs. One manufacturer of transmission-line towers 1 claims that the old style anchor, with the bottom diagonal of the tower attached to the anchor stub above ground, does not afford proper resistance to the horizontal force at the ground level and some- times leads to failure of the tower legs in the bottom panel. This manufacturing company uses a design of anchor in which the joint between the tower leg and the bottom diagonal is below the ground surface. Another point of importance is the surface of the footing in contact with the earth immediately above it. The weight of the 1 The American Bridge Company. TRANSMISSION-LINE SUPPORTS 347 cone of earth to be lifted may be ample to provide the desired factor of safety; but movement of the tower foundations may occur through the packing of the earth due to excessive unit pressure over the upper surface of the footing, and this movement may be appreciable notwithstanding that there may be no dis- turbance of the ground surface. A surface of not less than 1 sq. ft. for every 10,000 Ib. of the vertical force which will pull out the anchors should be provided, unless the nature of the soil is such as to justify a reduction of this allowance. 168. Concluding Remarks Regarding Steel Tower Design. Generally speaking, there is a tendency to economize in the cost of steel towers by using sections of structural steel in which stiffness is obtained by making the thickness of metal small in proportion to the other dimensions of the cross-section. It is true that light weight of parts and of the complete tower are important if the advantage of lightness can be obtained without sacrifice of other advantages, the chief of which is durability. When a transmission line is not intended to last longer than 15 or 20 years, these light sections are permissible; but for the more important and costly lines, it is well to avoid the use of metal thinner than ^ in. for the main members, or than %6 in. for the secondary or bracing members. In the writer's opinion it is not wise to use 4" by 4" angles for the corner legs less than %6 in. thick, although a thickness of ^ in. is not uncommon in towers actually in use at the present day. The ultimate life of such towers is, however, as yet unknown. Towers made of few pieces of comparatively heavy section steel will probably Drove more durable than those built of a larger number of lighter parts. ( If the temptation to use very light sections of structural steel is avoided, and if towers are regularly inspected and painted when and where necessary, their life should be 50 years or more. Galvanized towers are usually not painted; but it is not safe to rely upon the thin coating of zinc to prevent corrosion for more than a few years at or near the ground level. A casing of con- crete extending about 12 in. above ground level will afford protection; or the parts that are buried may be painted instead of being galvanized, and if the anchor stubs are made in two lengths, the upper length can at any time be replaced without interrupting the service. When considering designs of towers for along transmission line, 348 ELECTRIC POWER TRANSMISSION it is well to avoid if possible a number of different types, and where it is not necessary to increase the height, it may sometimes be found more economical to use two standard towers close together for supporting special long spans, or for turning sharp corners, than to design special towers for the purpose. An angle not ex- ceeding 7 degrees can usually be turned on a standard tower. This angle may even be as great as 10 degrees, especially if the length of the approach spans is decreased. In fact by reducing the length of approach spans, very much sharper angles can be turned; but it then becomes a question whether or not a special structure might not be the cheaper alternative. There is an unexplained prejudice against the guying of steel towers where extra strength to resist lateral loads is required. By giving proper attention to the method of guying, and inspecting the line at regular intervals, there is no apparent reason why this fairly obvious device to save the extra cost of special structures should not prove entirely satisfactory. It is true that, with the so-called rigid design of tower, a very small deflection at the point of attachment of the guy wire may be sufficient to produce permanent deformation of the structure, and there is a possibility that the tower may collapse under excessive load before the guy wires have taken up their proper share of the abnormal stresses. This is especially likely to occur if the towers are set on concrete foundations. On the other hand it is not impossible to design towers of the square base type with foundations purposely arranged to yield slightly; and if these structures are provided with guys (say of plow steel cables) fixed very securely to practi- cally unyielding concrete anchorages, it is probable that econo- mies might, in many instances, be effected. Guying of corner poles or of occasional poles on a straight run, when the more flexible type of "A "-frame structure is used, is generally to be recommended. A brief specification for a complete transmission line using steel towers is given in Appendix III. This line is generally simi- lar to the one for which an estimate of cost was given in Chapter III. 169. Determining Position of Supports on Uneven Ground. The lowest point of the span is not necessarily the point at which the wires come closest to the ground. When there is doubt as to the proper location of the supports in rough country, the method illustrated in Fig. 131, and described by Mr. J. S. Viehe TRANSMISSION-LINE SUPPORTS 349 in the Electrical World of June 15, 1911, will be found very con- venient. The curve a is the parabola corresponding to the re- quired tension in the particular wire to be used. The ratio of the scale of feet for vertical measurements to the scale for horizontal measurements should be about 10 to 1. The dotted curves 6 and c are exactly similar to a, but the vertical distance ab repre- sents the minimum allowable clearance between conductor and ground, while the vertical distance ac is the height above ground level of the point of attachment of the lowest wires to the stand- ard transmission pole or tower. These curves should be drawn on transparent paper: they can then be moved about over a FIG. 131. Method of locating position of towers in rough country. profile of the ground to be spanned, drawn to the same scale as the curves, until the best location for the supports is found. The point P where the curve 6 touches the ground line is seen to be far removed from the lowest point of the parabola, in the example illustrated in Fig. 131. A little practice will make the finding of the points A and B an easy matter, even if the length of span, or distance between A and B, must be kept between close limits. This method is particularly applicable to long-span lines carried over rough country. 170. Study of Deflections and Stresses in Flexible Tower Lines. Consider a series of poles as in Fig. 132, the end one being rigid while all the others are flexible and of equal height and stiff- ness. It is assumed that all spans were originally of equal length I, and that there were 6 wires in each span, strung to a 350 ELECTRIC POWER TRANSMISSION tension of S pounds per square inch and having a corresponding sag s. In span No. 1, terminating at the rigid support, some of the wires have been severed, leaving only a wires in this span. It is assumed also that there is no slipping of the wires in the ties on the pin type insulators, and no yielding of pole foundations. The elastic deflection of a pole or tower considered as a beam fixed at one end and loaded at the other is PH* ZMI where P is the load, H the height, M the elastic modulus, and 7 the moment of inertia of the cross-section. In the special case considered, the value of P, which produces the deflection 5i of the first flexible pole, is P = A(bS, - aSi) b Wires _ I u u u FIG. 132. Flexible pole line. where A is the cross-section of one conductor and Si and S% are the stresses in the conductors of spans No. 1 and No. 2 respect- ively. It is assumed that all the wires are attached to the pole tops at a point H in. above ground level. By putting K =5-577' the successive deflections maybe written: 81 = KA(bS 2 - aSi) (164) 6 2 = KAb(S - S 2 ) (165) and the sum of the deflection of a series of flexible poles of the same height and stiffness is A = KA(bS n - oSi) (166) where n is the number of the last span. It is usually safe to as- sume that S n is equal to the initial tension S in the fourth or fifth span from the break. TRANSMISSION-LINE SUPPORTS 351 Fig. 133 shows the conductors in the first span with a sag s under normal conditions with b wires in the span, and a smaller sag Si after some of the wires have been cut, leaving only a wires in the span. For simplicity in calculating the movement of the point of attachment of the wires on the flexible pole, instead of considering the span as increasing in length from I to (Z + 5), the span I may be supposed to remain unaltered while the length of the conductor is reduced by pulling it through the tie of the insulator (G) on the flexible pole until the sag is reduced from s to s\. The length of wire pulled through in this manner may, for all practical purposes, be considered equal to the actual pole- top deflection, 8. This assumption is justifiable since the de- flection 8 is always small relatively to the span I. The length of the (parabolic) arc with sag s is FIG. 133. Elongation of wire in span due to deflection of pole top. and with sag Si The difference is _ _ 8(s 2 si 2 ) ~37~ to which must be added the elongation due to the stretch of the wire under increased tension; this is Xi (Si - S} M ~ or, with quite sufficient closeness, l(Si S) ~W~ 352 ELECTRIC POWER TRANSMISSION Hence the deflection of the first flexible pole expressed in terms of the sag and tension of the conductors in the first span is : (167) Returning again to the arrangement of line depicted in Fig. 132, we shall consider (1) the total pull of all the wires in span No. 2 and the effect of this pull on the first flexible pole if all the wires are broken in span No. 1, and (2) the effect on the first flexible pole and the stresses in the remaining wires in No. 1 span on the assumption that all the wires in this span are not broken. When the particulars of the poles are known, so that the factor K in the formulas for deflection can be determined, it is desired to calculate the stresses in poles and wires corresponding to the new conditions of equilibrium; or, if the poles have yet to be designed, the factor K must be determined, in order that the stiffness of the poles shall satisfy certain necessary or assumed conditions, such as the maximum deflection of pole top which will not stress the remaining wires in span No. 1 beyond the elastic limit of the conductor material. (A factor of safety must be used to allow of momentary increased stresses due to probable mechanical surges.) 171. Numerical Example: Transmission Line with Flexible Supports. No attempt will be made to obtain an exact mathe- matical solution of these problems, but close approximations can be obtained with sufficient accuracy for practical purposes, espe- cially when it is considered that many possible influencing factors, such as the yielding of foundations and the slipping of wires in the ties, cannot be taken into account even in the most complete mathematical treatment of the subject. It is assumed that the poles are equidistant and in a straight line, and that the first support is rigid, all as indicated in Fig. 132. Four separate limiting conditions will be considered : (A) All wires are severed in the first span, and the pole between spans 2 and 3 is considered to be rigid. (B) All wires are severed in the first span, but the pole between spans 2 and 3, and all subsequent poles, are considered to offer no resistance to deflection in the direction of the line. (C) There are a wires remaining in span No. 1, and 6 wires in all other spans. The pole between spans 2 and 3 is considered to be rigid. TRANSMISSION-LINE SUPPORTS 353 (D) There are a wires in the first span, but the pole between spans 2 and '3, and all subsequent poles, are considered to be infinitely flexible. The transmission line will be supposed to have the following characteristics: Six No. 2-0 aluminum conductors. Cross-section of conductor, A = 0.1046 sq. in. Length of span, I = 400 ft. Normal sag = 9.76 ft., which corresponds to Stress S = 2400 Ib. per square inch. It is assumed that there is no grounded guard wire above the conductors, and that the average height of the point of attach- ment of the wires above ground level is H = 45 ft. The modulus of elasticity of aluminum cables for the purpose of these calculations is assumed to be M = 7,500,000. The flexi- ble towers are in the form of braced "A"-frames, each vertical limb consisting of one 7-in. steel channel of light section (9% Ib. per foot) . The moment of inertia of the section of such a channel is 21.1, and since there are two channels, the value of 7 is 21.1 X 2 42 2 = 42.2 and the section modulus Z= ~r = (say) 12. The elastic modulus for steel is M = 29 X 10 6 . The factor K for use in pole deflection formulas as previously given is therefore _ (45 X 12)3 _ ~ 3 X 29 X 10 X 42.2 ~ The maximum deflection of this particular structure before permanent deformation would take place will occur when the difference of pull due to the wires is such as to stress the metal to (say) 30,000 Ib. per square inch. The resisting moment is S X Z = 30,000 X 12 and the resultant pull at the pole top will The maximum allowable deflection is therefore, 5= K X 667 = 0.0428 X 667 = 28.5 inches. Case (A). All wires broken in span No. 1; second pole beyond break considered rigid. 354 ELECTRIC POWER TRANSMISSION Since all the wires are severed in span No. 1 (a = 0) it is not possible to make use of formula (167), but a similar formula can be used which expresses the deflection in terms of the constants for span No. 2. This formula is .8 (168) By calculating 61 for various arbitrary values of 2 smaller than S, curve No. 1 of Fig. 134 can readily be drawn. This gives the relation between the stress S z in the wires of the second span and the pole-top deflection 5i on the assumption that the second pole beyond the break is rigid. On the same diagram draw the 200 1400 1600 1800 2000 2200 Stress in wires of span No.2 = Sz Jbs. .per.sq. inch FIG. 134. Graphic solution of Problems (A) and (B). 2400 straight line marked curve No. 2, which gives the relation between pole deflection and the stress $2, as given by formula (164) when the tension S : in wires of the first span is equal to zero. The point of crossing of curves No. 1 and No. 2 evidently indicates the deflection corresponding to the condition of equilibrium. This deflection is Si = 29.5 in. and stress S 2 = 1100. It will be noted that in this particular example the deflection is about the same as the maximum permissible deflection (28.5) previously calculated ; but even if allowance be made for shocks and mechanical surges, it is probable that the pole would not suffer serious injury, because some of the wires would be liable to TRANSMISSION-LINE SUPPORTS 355 slip in the ties and so relieve the tension. If wind pressures acting on snow or ice deposits are added to the stresses due to weight of conductor material only, the strain will be greater, but on the other hand, much sleet deposit is liable to be shaken off the wires in the event of a sudden severing of all the wires in the first span. The above results are, however, based on the assumption that the second pole beyond the break is rigid, which may not be in accordance with practical conditions. Case (B). Conditions as above; but the second and subse- quent poles beyond the break are supposed to be infinitely flexible (K = ). In this case the tension S z will not depend upon the deflection of the first flexible pole; it will be equal to the original tension S = 2400 for all values of the deflection Si. The deflection ob- tained when S z = 2400 is of course readily calculated by means of formula (164), or it can be read off Fig. 134, since it is the deflection indicated at the point where curve No. 2 meets the vertical ordinate for Sz = 2400. This value of 1 is 64.5 in., which would lead to permanent deformation of the flexible structure. The actual deflection of the first pole in a series of flexible poles of equal stiffness would lie somewhere between these limiting values of 29.5 in. and 64.5 in. if the law of elasticity may be considered to apply in the case of the higher deflections. As a general rule the breaking of all wires hi one span will lead to the wrecking or permanent deflection or uprooting of the first pole, which cannot be at the same time flexible enough greatly to reduce the combined pull of all wires in span No. 2, and yet strong enough to resist the ultimate combined pull of these wires. There would be an exception in the case of short spans with tall flexible poles; and in any case it is probable that only the first pole would be damaged or moved in its foundations. It is rare that all the wires in one span are broken simulta- neously unless the design of the line is such that the severing of one or more wires leads necessarily to the rupture of the remaining wires owing to the excessive stresses imposed on them. The cal- culation of stresses and deflections when a certain number of wires remain in the faulty span is more difficult than in the cases already considered, but the solution is of greater practical value. Case (C). There are a wires in the faulty span and 6 wires in the sound spans. The second pole beyond break is considered 356 ELECTRIC POWER TRANSMISSION rigid. (For the purpose of working out numerical examples it will be assumed that only one wire remains in faulty span; thus a = 1 and b = 6.) Instead of only two equations, there are now three equations to be satisfied simultaneously; these are: (a) Formula (164): 81 = KA (bS 2 - aSi) = 0.0269S 2 - 0.00448Si (b) Formula (167), giving deflection in terms of elongation of remaining wires in span No. 1: (c) Formula (168), giving deflection in terms of the shortening of the wires in span No. 2. (This relation is given by curve No. 1 already plotted in Fig. 134.) It should be mentioned in connection with formulas (167) and (168) that, by assuming a constant length of span, the sag s is always inversely proportional to the stress S. The assump- tion of a constant length of span for the purpose of simplifying the relation between sag and tension introduces no appreciable error in practical calculations. In the particular example from which the curves are plotted, and the numerical results obtained, 23,420 the relation is s = ~ o Proceed, now, to plot curve No. 3 in Fig. 135 from formula (167) by assuming various arbitrary values of Si from the lowest possible limit of Si = S = 2400 up to the elastic limit of about 13,000. For a reason to be made clear hereafter this curve should be drawn on transparent paper; the horizontal scale used for the values of Si may be arbitrarily chosen, but the scale of ordinates giving the deflections 5i must be exactly the same as used for Fig. 134. On the same diagram (Fig. 135) draw also the straight line marked curve No. 4, giving the relation between Si and the quantity KAaSi. This latter quantity when subtracted from the quantity KAbS 2 will give the pole deflection to fulfil the condition of formula (164). The reason for drawing the curves of Fig. 4 on transparent paper will now be clear. The transparent paper with the curves of Fig. 135 is placed over Fig. 134, with the horizontal datum lines of zero deflection TRANSMISSION-LINE SUPPORTS 357 coinciding as shown in Fig. 136. The point of intersection of curves No. 1 and No. 3 will give the corresponding values of the stresses Si and S z l but with a pole having definite elastic prop- I- 10 o 10000 12000 Stress in wires of span No.l = S\ Ibs. per uq. inch FIG. 135. Curves to be drawn on tracing paper for solution of Problems (C) and (D). Lower Sheet with Curves Nos.l and 2 Transparent Paper with Curves Nos.3 and 4 FIG. 136. Graphic solution of Problems (C) and (D). 1 There is a definite value of Si for any given value of 2 independent of all considerations of pole stiffness and size of wire and number of wires in adjoining spans. This is the relation which will satisfy formulas (167) and (168) simultaneously; it is expressed by the equation (2s* - 8l - 358 ELECTRIC POWER TRANSMISSION erties there is only one value of the deflection which will satisfy the three conditions previously referred to. The deflection as a function of the pole stiffness is the distance EF (Fig. 136), being the difference between the corresponding ordinates of curves No. 2 and No. 4. By moving the tracing paper with the curves No. 3 and No. 4 over the other curves until the distances HG and FE on the same vertical ordinate are equal, -the deflection corresponding to the condition of equilibrium is readily obtained. If preferred, the curve OPRE, representing the sum of the quantities of curves No. 3 and No. 4, may be drawn on the tracing paper instead of the curve 4, and when the point of intersection (E) of this new curve with curve No. 2 on the lower sheet lies on the same vertical ordinate as the junction ((7) of the curves No. 1 and No. 3, the distance HG will be the required deflection. The solution of the numerical example worked out in this manner is 81 = 10.2 in. 51 = 7400 Ib. per square inch. 5 2 = 1500 Ib. per square inch. Case (D). Same conditions, with the exception that the second pole beyond the break, instead of being rigid, is assumed to be infinitely flexible. This assumption is made also in the case of all subsequent poles. This means that 2 = S = 2400 whatever may be the amount of deflection of the first flexible pole, and the problem can be solved graphically as indicated above, the only difference being that curve No. 1 giving the relation between 81 and $2 when the second pole beyond the break is rigid must be replaced by the vertical line SW (Fig. 136), being the ordinate corresponding to a tension S 2 = 2400. The numerical solution in this case is: di = 13.2 in. Si = ll,4501b. persq. in. It is interesting to note that there is little difference between the deflections for the two extreme cases (C) and (Z>); the average value for 5i is 11.7 in., corresponding to a stress Si = 9400 in the remaining wire of the faulty span. This is well below the elastic limit, and it is probable that this wire would not break even if the five other wires were severed. The figures chosen for illustrating TRANSMISSION-LINE SUPPORTS 359 the calculations -relate to a practical transmission line, and it will be seen that the stresses and deflections corresponding to the state of equilibrium after the severing of one or more wires in one span can, with the help of simple diagrams, be predetermined within reasonably narrow limits. 172. Erection of Steel Tower Transmission Lines. This book is not intended to give practical advice to construction engineers or the men actually engaged in the work of erecting poles or towers and stringing wires. A competent construction engineer with experience in handling men and materials in the field, should be given a free hand in planning and executing the work of erecting a power transmission line; and such a man will not derive much assistance from books. On the other hand, there are some excel- lent books available dealing with the more practical side of trans- mission line engineering. These include the various electrical engineering handbooks. The reader desiring information on the methods ordinarily adopted in carrying out the details of con- struction, is referred to these other sources of information; also to the papers and articles which appear from time to time in the Journals of the engineering societies and in the technical press. Appendices II and III which follow this Chapter also contain items of some practical interest connected with the setting out and erection of wood pole and steel tower lines. The principal reason for referring to these matters in this place is to emphasize the importance of devoting much time and thought to the various details of overhead line construction before the work is actually started. The proper setting out of the line is among the most important matters connected with over- head construction. If a line is not carefully surveyed and planned in every detail, it will often be impossible to get good and reliable service from it. This does not mean that the commer- cial aspect of the undertaking is not of prune importance; on the contrary, it is the only aspect from which an engineering under- taking of the kind under consideration should be viewed. But this is not equivalent to saying that a small first cost is always desirable, or that even a short low-voltage transmission line can be constructed and operated economically by persons without engineering skill and experience. It is an easy matter to find examples of lines that have cost too much; but it is not impossible to find the transmission line that has cost too little in the first instance. 360 ELECTRIC POWER TRANSMISSION Generally speaking, the writer believes that not enough atten- tion is paid to preliminary investigations and estimates of power transmission lines. The construction of comparatively short lines for moderate voltages appears to be, and indeed is, a fairly simple piece of work; yet in respect to economy and service such lines may be a source of endless trouble if they have been planned and constructed without regard to the fundamental principles of engineering. APPENDIX I INDUCTANCE OF TRANSMISSION LINES WITH ANY ARRANGEMENT OF PARALLEL CONDUCTORS 1 The manner in which the inductance and the induced e.m.f. can be calculated when the conductors of a three-phase system occupy the vertices of an equilateral triangle, was explained in Chapter II; and it was also stated that a departure from the symmetrical arrangement of conductors does not modify the calculated results to a great extent. It will be interesting to study the problem in its broader aspect, with a view to ascer- taining what is the nature and magnitude of the modifying fac- tors. It is proposed to indicate a simple method of calculating the total induced e.m.f. in any conductor of an electric-energy transmission system, whatever may be the actual arrangement or relative positions of the conductors. It is assumed in all cases that the conductors are of circular section and that they remain parallel with each other throughout the whole distance of transmission. Calculation of Total Resultant Flux Surrounding One Con- ductor When There Are Several Return Conductors. In Fig. 1 the total outgoing current / is supposed to flow along one con- ductor, while the total return current is divided between a number of conductors, the condition being that 7 = -(A + /2 + /3 +/n) Let di, d z , d 3 , etc., represent the distances between centers of the corresponding conductors carrying the return currents and the conductor carrying the outgoing current, and note that the total flux surrounding the latter conductor may be considered as the algebraic sum of several separate fluxes, namely, the flux due to a current I\ returning at a distance d\] the flux due to a current 7 2 returning at a distance d z , and so on, for any number of components of the total current 7. All these separate compo- nents of the total flux can readily be calculated by means of formula (25) of Article 43, Chapter IV, and the expression for the total flux surrounding a conductor in which the current 7 returns 1 This Appendix is a reprint, with slight changes and omissions, of articles which were first published in the Electrical World of May 23, 1908 and Sept. 15, 1910. 361 362 ELECTRIC POWER TRANSMISSION along a number of separate conductors, as indicated in Fig. 1, becomes : 2Z T r i di T , d z r , d n ~\ ,.. * - jjjl"- * gt 7 2 ge 7 ..... ~ n ge 7J ( } where r stands for the radius of cross-section of the conductor carrying what will be thought of as the outgoing current /. In the case of energy transmission by polyphase currents, with any number of conductors, the algebraic sum of the currents in the conductors must, at any given instant, be equal to zero. Any one conductor may be looked upon as carrying the outgoing current, while the remaining conductors together carry the return current. Formula (1) can, therefore, be used for calculating the effective flux of induction surrounding any one conductor in a polyphase transmission, whatever may be the arrangement of the con- ductors. The phase relations of the various component fluxes must, how- ever, be taken into account, and for FIG. 1. Section through ,. , i-i ITJ- r four parallel conductors. this reason the graphical addition ot vector quantities with the help of a diagram will be found most convenient. Instead of drawing the vectors representing magnetic flux components in phase with the current vectors the component vectors of the resulting e.m.f. of self-induction may be drawn in this case 90 time-de- grees behind the corresponding current vectors. Calculation of E.M.F. of Self- and Mutual Induction. In order to calculate the induced e.m.f. it will be advisable first to put equation (1) in a more practical form. The symbols /i, 1 2, etc., in equation (1), when the latter is to be used for calculating the maximum value of the induction due to an alternating cur- rent, must be considered as representing the maximum value of the current wave; but it will be more convenient to assume sinusoidal currents, and then let these symbols stand for the virtual (or r.m.s.) value of the currents. The procedure is now as indicated in Article 45 of Chapter IV, leading up to formula (28) which may be written: Reactive volts per mile | _ , d . , T /m ,. , = al log - + bl (2) of single conductor j r where a = 0.00466/ and b = 0.000506/ INDUCTANCE OF TRANSMISSION LINES 363 The item bl is ihe reactive voltage component due to the flux set up inside the material of the conductor by the current /. The flux producing this increased reactive e.m.f. is not included in the flux as calculated by formula (1). It may generally be neglected in calculations of high voltage overhead transmission lines. The final expression for the reactive voltage drop per mile of conductor when there are several parallel return conductors is: E = a [- I, log - 7 2 logy 2 . . . . - I n logy"] + 67 (3) Numerical Example. Three-phase Transmission. Consider the special case, which not infrequently arises in practice, of the conductors of a three-phase transmission being arranged as indi- cated in Fig. 2 that is, with the centers of the three conductors lying in the same plane, the minimum distance, d, between any two of the wires being approximately equal to the side of the equilateral triangle which would have * ^ _3 been adopted had the triangular ar- U d H d *^2r rangement been decided upon. FIG. 2.-Three conductors in In a three-phase transmission system one plane. the current flowing out through any one wire may, as previously mentioned, be considered as returning along the two remaining wires, and when the three conductors occupy the vertices of an equilateral triangle the whole of the return current is at a distance d from the out- going current. This condition also applies to the middle con- ductor (No. 2) in the arrangement shown in Fig. 2; but it does not apply to either of the outside conductors, Nos. 1 and 3. In the case of conductor No. 1 a part of the outgoing current returns along conductor No. 2 at a distance d, while the remainder returns along conductor No. 3 at a distance 2d', so that the total flux of induction surrounding conductor No. 1 must necessarily be greater than that surrounding conductor No. 2. The same argument applies to conductor No. 3. Applying formula (3) to the arrangement of conductors, as shown in Fig. 2, the quantity between brackets in the case of conductor No. 1 becomes: d , , 2d - 1 2 log - - 7 3 log = - (7 2 + 7 3 )log^ -7 3 log2 364 ELECTRIC POWER TRANSMISSION The total induced e.m.f . per mile of conductor No. 1 will therefore be: E l = a X [/i log ^ - 7 3 log 2] + 67 j (4) Similarly, for conductor No. 3: E 3 = a X [/ 3 log * - 7i log 2] + b/ 3 (5) while the volts induced in the middle conductor (No. 2) will be simply: E 2 = a X 1 2 log - + &7 2 (6) It is interesting to note that what may be referred to as the disturbing element in the case of the two outside wires (the quantities 7 3 log 2 and I\ log 2 respectively) is not dependent upon the actual diameter or distance apart of the conductors. It consists of an e.m.f. component either 30 time-degrees or 150 time-degrees behind the phase of the line current, depending upon the order of the phase rotation; and the magnitude of this e.m.f. component relatively to the total e.m.f. of self-induction will depend upon the value of the ratio -. If d is large and r relatively small, as in the case of a high-pressure overhead transmission system, then the first quantity between brackets, in equations (4) and (5), is relatively large, and the disturbing element (/ 3 log 2 or 7i log 2) is usually negligible. On the other hand, if the conductors consist of three separate single cables, laid side by side in a trench, with the distance, d, between them small in comparison with the diameter, 2r, of the cables, then the "disturbing element" becomes of greater importance relatively to the total induced e.m.f. In order to form some idea of the magnitude of this out-of- balance component of the induction, it will be well to work out two numerical examples, one for a high-tension overhead scheme and the other for a low-tension transmission system with the three conductors in comparatively close proximity. Example 1. Assumed data: Three-phase power transmitted = 20,000 kw.; e.m.f. = 110,000 volts; power-factor = 0.8; fre- quency / = 25 cycles per second; length of line = 200 miles. Conductors of aluminum; diameter, 2r = 0.6 in. Minimum dis- INDUCTANCE OF TRANSMISSION LINES 365 tance between -wires, d = 10 ft. = 120 in. On the above data the current per conductor is about 130 amp. With the aid of formulas (4), (5) and (6) it is an easy matter to determine the induced e.m.fs. in the several conductors, and since the quantity, log - = log TT^- = 2.6021, while log 2 = 0.3010, it will at once be T U.o seen that the "disturbing element" is relatively small. The e.m.fs. induced in each conductor 200 miles long, in round figures (neglecting the component 67 due to the internal flux) are as follows: In the middle conductor (No. 2), 8000 volts, the time-phase of which is exactly one-quarter cycle behind the time-phase of the current /2. In conductor No. 1, an e.m.f. component of 8000 volts, exactly a quarter cycle behind the current /i less another component (referred to as the disturbing element) equal to about 915 volts, the phase of which is exactly one-quarter cycle behind the current 1$. The resultant is the difference between two vector quantities separated by a time-phase angle of 120 deg., so that this resultant is actually greater than either of the two compo- nents, as will be shown hereafter. In conductor No. 3 there will be an e.m.f. component of 8000 volts, one-quarter cycle behind the current /a, and a component of 915 volts, one-quarter cycle behind /i. Example 2. Assumed data: Three-phase power transmitted = 20 kw.; e.m.f. = 110 volts; power-factor = 0.8; frequency / = 60 cycles per second ; current per wire = 130 amp. ; distance of transmission = % mile; three single cables in trench, lying in the same plane with a distance between centers d = 3 in.; diameter over copper = 2r = 0.5 in. In this example the quantity log - 3 = 1.0792 The ratio between log 2 and this number is = - 28 - is to say, the component of the total induced e.m.f., which appears only in the two outside conductors, as indicated by formulas (4) and (5) is, in this example, numerically greater than a quarter of the more important component; while in the previous example of a high-tension overhead transmission system the ratio was 366 ELECTRIC POWER TRANSMISSION 0.115, being considerably smaller because of the greater 0.30JO 2.6021 distance between the wires. Vector Diagram Illustrating Example 2. The vectors I\, 2z and 7 3 in Fig. 3 represent the currents in the three conductors, the time-phase angle between them being 120 deg. The rotation of the phases is assumed to be in the order 7i, 7 2 , Is', in other words, 7 2 lags behind 7i by one-third of a cycle, and 7 3 lags behind 7 2 also by one-third of a cycle. The lengths of these vectors are such as to represent the line current of 130 amp.; but, as the diagram has been drawn to illustrate the phase angles and magni- tudes of the various components of the induced e.m.fs., the magni- tude of the current vectors need not be considered. If the numer- ical values of the induced volts are determined with the aid of formulas (4), (5) and (6), it will be found that the component common to all three conductors amounts to 21.6 volts, while the "disturbing element" that is, the component appearing in the two outer conductors only amounts to 5.5 volts. The vectors OB, OV Z and OD must, therefore, be drawn of such a length as to represent 21.6 volts in a direction exactly 90 time-degrees behind the cor- responding current vectors; and, so far as the middle conductor is concerned, the vector OV 2 will represent the whole of the in- duced e.m.f.; but in the case of conductor No. 1 (carrying current 7i), OA must be drawn exactly 90 time-degrees in advance of 07 3 that is, exactly opposite to 07), because of the negative sign in equation (4) and of such a length as to represent 5.5 volts. By combining OA with OB in the usual way, OVi is obtained as representing the total e.m.f. induced in conductor No. 1. In a similar manner OV 3 is obtained for the total induced e.m.f. in conductor No. 3. It is interesting to note that OVi lags be- hind the current 7i by a time interval greater than a quarter period, while the lag of the induced volts F 3 behind the current 7 3 is less than a quarter period. B FIG. 3. Vector diagram. Three conductors in same plane. INDUCTANCE OF TRANSMISSION LINES 367 In the particular example under consideration the calculated value of Fi or F 3 is 24.8 volts; F 2 being 21.6 volts. It is not difficult to understand why the magnitude and phase relations of the induced e.m.fs. in the various conductors of a polyphase transmission are not the same for an unsymmetrical arrangement of conductors as for an arrangement in which each conductor is similarly placed in relation to all the other conduc- tors. With an unsymmetrical arrangement, the unbalancing effect may be said to be due to the mutual induction between the loops formed by different pairs of wires; there may, in fact, be a transfer of energy between one loop and another just as in the case of the primary and second- ary windings of a transformer. Effect of Transposing the Conductors. If each conductor of the arrangement referred to in the above example is made to occupy, in turn, the position midway between the remaining two conductors for a distance FIG. 4. Vector diagram illustrat- ing effect of transposing conductors lying in the same plane. equal to one-third of the total distance of transmission, it is obvious that the out-of-balance effect will be corrected. It will, however, be of interest to as- certain what will be the numeri- cal value of the (equal) voltages induced in the three conductors if transposed in the manner suggested. It is not necessary to consider more than one of the conductors, and, in Fig. 4, OB repre- sents (as in Fig. 3) that portion of the e.m.f. induced in conductor No. 1 which remains unaltered whether the conductor be midway between the other two, or be itself one of the outer conductors. The length of this vector will, therefore, be such as to represent 21.6 volts. Now, when the arrangement of the conductors is in the order 1,2,3 (as in Fig. 2), the "disturbing element " will be BG, drawn 90 degrees in advance of OI 3 , exactly as OA (or BVi) in Fig. 3 ; but the length of this vector, instead of being equivalent to 5.5 volts, will be only one-third of this value, or 1.83 volts, because conductor No. 1 occupies this position over one-third of the total distance of transmission. When the arrangement of the conductors is 1, 3, 2, the "disturbing element" will be GVi 368 ELECTRIC POWER TRANSMISSION (Fig. 4), drawn 90 degrees in advance of 07 2 . Clearly BGVi is an equilateral triangle, and the resultant of the induced e.m.f. in conductor No. 1 is OVi, drawn 90 time-degrees behind the current vector 01 1 and equal in magnitude to the algebraic sum of OB = 21.6 volts and BVi = one-third of 5.5 volts. If, therefore, the wires of a transmission line are disposed in one plane, as indicated in Fig. 2, but transposed at intervals so that each wire shall occupy the middle position over a space equal to one-third of the distance of transmission, then the resultant induced e.m.f. per conductor will, so far as phase is concerned, lag behind the current by a quarter period, exactly as if the wires occupied the vertices of an equilateral triangle; but the amount of the induced volts will be somewhat greater than in the latter case, under otherwise similar conditions. The numerical value of the induced volts per conductor that is, the length of the vector OVi in Fig. 4 can be calculated by the formula: W (7) where / is the current in any one conductor, and the two quan- tities between brackets have merely to be added algebraically. If preferred the quantity between brackets may be written: or 7 log (l.26 J, so that formula (7) appears in the form: E = 0.00466/7 log ^^ + 0.000506/7 (8) Inductance of Electric Transmission Lines as Affected by the Subdivision of the Circuits and the Arrangement of the Conductors. There are reasons in favor of transmitting large amounts of electric power through two or more sets of wires, quite distinct from mechanical considerations or the increased security against a total shut-down in the event of accidents. The inductive drop of pressure may be reduced by substituting, for a single set of transmission lines, two or more sets of suitably arranged lines of a correspondingly reduced cross-sectional area. Whether or not the subdivision of a transmission line into two or more parallel circuits would be justifiable in practice will depend upon economic and other considerations which it is not proposed to touch upon here. INDUCTANCE OF TRANSMISSION LINES 369 Single-phase Systems. In Fig. 5 the two conductors of a single-phase transmission are shown, with distance d between centers of wires. The current may be considered as going out through the conductor 1 and returning through conductor 2. The diameter of the wire is assumed to be 2r and the current / amp. The formula which gives the induced volts per mile of single conductor when the whole of the current may be considered as returning at a distance d from the center of the outgoing con- ductor is (9) where a has the value given above, and the item 67 has been omitted as it is not necessary to include it when considering differences of reactive e.m.fs., especially in the case of overhead systems where its magnitude is relatively small, and frequently negligible. This formula alone is sufficient to indicate that an improvement in the matter of indue- tive voltage drop is to be expected if, in- stead of transmitting the total current / through one pair of conductors, there be provided two or three pairs of conductors spaced sufficiently far apart to prevent mutual inductive effects, each pair being of sufficient cross-section to carry one-half or one-third of the total current, as the case may be; be- cause, although the quantity log- will increase slightly on account of the reduction in the dimension r, this increase will not be of nearly so much importance as the reduction of /. Numerical Example. In order to illustrate the above point a few examples will be worked out based on the following as- sumed data: Total current, / = 100 amp. Diameter of single conductor to transmit the total current, 2r = 0.5 in. Frequency, / = 60 cycles, from which a = 0.279. Distance between centers of wires (corresponding to a pres- sure of about 50,000 volts), d = 70 in. If the transmission line is divided into two equal sections, the current in each section will be 50 amp., and for equal total weight of copper (leading to the same ohmic drop of pressure), 24 370 ELECTRIC POWER TRANSMISSION the radius of each conductor will be r -r- \/2. Similarly, if there are three equal sections, the current will be 33.33 amp., and the radius of the conductors r -4- \/3. The induced volts as given by formula (9) work out as follows for the three conditions : Single pair of lines e = 68.34 volts (10) Two pair of lines of equal total cross-section, e = 36.25 (11) Three pair of lines of equal total cross-section, e = 25.00 (12) These figures show that the inductive drop of pressure on a single-phase transmission may be reduced by splitting up the current and transmitting along two or more pairs of lines spaced sufficiently far apart to prevent appreciable magnetic interfer- ence between the sets of lines; and the reduction of the inductive drop is very nearly in proportion to the number of subdivisions of the single line. Although electric transmission systems have been arranged with two distinct sets of conductors run upon separate pole lines spaced sufficiently far apart to avoid magnetic interfer- ence, such an arrangement is necessarily costly. Consider, therefore, two alternative arrangements, shown in Figs. 6 and 7, by which a single circuit can be split up into two parallel cir- cuits, the four wires being carried on the one set of poles with the spacing between the individual wires as small as possible that is, such that in no case shall the distance d between out- going and return conductors be less than the minimum deter- mined by the voltage of the supply. In Fig. 6 is shown a symmetrical arrangement with the four conductors of equal cross-section occupying the corners of a square; the outgoing conductors are marked 1 and 3, and the return conductors, 2 and 4. Even if the two circuits 1-2 and 3-4 are connected in parallel at both ends of the line, the sym- metry of the arrangement will insure that the total current will divide itself equally between the two sets of conductors. The effective or resultant magnetic flux surrounding any one conductor will, for the same reason, be equal to that which INDUCTANCE OF TRANSMISSION LINES 371 surrounds any one of the remaining three conductors. It will, therefore, suffice to calculate the e.m.f. of self-induction gen- erated in any one conductor. Consider the conductor 1, in which there is the current ~- If the other outgoing conductor, 3, were situated anywhere on the dotted circle of radius d, passing through 2 and 4, then the magnetic effect of the current in 3 so far as conductor 1 is con- cerned would counteract the effect of the return current in either 2 or 4. On the basis of the data previously assumed, the flux around 1 would generate an e.m.f. of 36.25 volts, as in equa- tion (11). If, on the other hand, conductor 3 were coincident with 1, there would be the condition of the full current I in the conduc- -* *, v- -i FIG. 6. FIG. 7. FIGS. 6 and 7. Alternative arrangements of conductors. Single-phase transmission. tor 1, the whole of which would be returning at a distance d, and the induced volts would be 68.34, as given in equation (10). With the conductor 3 situated at a distance \/2d from conduc- tor 1, as shown in Fig. 6, the resultant effective flux surrounding conductor 1 may be considered as the difference between the flux due to a current I up to a distance d less the flux due to a current 1/2 up to a distance -\/2d; and this resultant flux would produce a back e.m.f. E~ai log -^fr - 4 log r^v72 < 13) On the data previously assumed, the e.m.f. is E = 72.5 - 38.39 = 34.11 volts. (14) Thus, by arranging the conductors of the divided circuit in the manner shown in Fig. 6, which permits of the four wires being 372 ELECTRIC POWER TRANSMISSION supported on the one set of poles, a better result is obtained in regard to inductive voltage drop than if the two circuits had been run entirely separately; the voltage drop in this latter case being 36.25, as in equation (11). If, on the other hand, the position of one pair of conductors be assumed to be reversed, as indicated in Fig. 7, then the magnetic flux in the loop formed by the outgoing and return conductors 2 and 3 has no effect on the conductors 1 and 4, and the effective flux surrounding any one conductor is clearly that due to a current ~ returning at a distance -\/2d: the in- duced volts per conductor will be 38.39, this being the value of the second term in formula (13). With an arrangement of conduc- tors, as in Fig. 7, it is obvious \^ that the conditions are worse than < % , if the two circuits are quite dis- ^Ax tinct, because a portion of the flux J -^ / 3' produced by one pair of conduc- ' tors, such as 3 and 4, passes also through the loop 1-2, thereby in- creasing the inductive drop in these wires. FIQ. 8. Arrangement of con- Three-phase Systems. The satis- ductors t h r e e - phase transmis- to inductive drop when a single- phase circuit is split up into two circuits arranged as indicated in Fig. 6, suggest that a somewhat similar arrangement might be adopted with advantage in the case of polyphase transmissions. An arrangement of wires suitable for three-phase transmission is shown in Fig. 8. Here the three-phase line is supposed to be split up into two parallel three-phase circuits, 1, 2, 3 and 1', 2', 3'. The arrangement being symmetrical and all conductors being assumed to be of equal size, the same amount of current will be carried by each of the six conductors, provided the load is a balanced one, such as is usual in the case of a three-wire, three-phase system. With the arrangement of wires as in Fig. 8 the minimum dis- tance d is maintained between all wires at different potentials, and the current in conductors, such as 1 and I', placed at opposite INDUCTANCE OF TRANSMISSION LINES 373 ends of a diameter, will be of the same time-phase and equal in magnitude. It will be interesting to work out a numerical example based on data already assumed in connection with the single-phase trans- mission, namely, a current of 100 amp. per phase and a mini- mum distance, d, of 70 in. between conductors at different poten- tials. The points to bear in mind are: (1) That owing to the symmetrical arrangement of the con- ductors, with the rotation of the phases always in the same direc- tion, the total effective magnetic flux around any one conductor is the same (except in regard to phase) as that which surrounds any one of the other five conductors. The calculations can therefore be made for any one conductor, such as No. 1. (2) That the current in any outgoing conductor, such as 1, may be considered as returning through the five remaining conductors, due attention being paid to phase relations. (3) That the resultant of the currents in conductors 2' and 3', or the resultant of the currents in conductors 2 and 3, is equivalent to a current equal to that in conductor 1, but of opposite phase. The total effective flux around conductor 1 may, therefore, be considered as the resultant of three component fluxes: (a) A flux due to a current ~ returning (through 2' 3') at a distance d; plus (b) a flux due to a current ^ returning (through 2-3) at a distance \/3d; less (c) a flux due to a current ^ re- turning (through 1') at a distance 2d. The numerical values for the induced volts are found to be: (a) = 36.25 [being the same as inequation (11)]; (6) = 39.47; (c) = 40.49; and (a) -f (b) - (c) = 35.23. If two separate three-phase lines spaced a considerable distance apart were substituted for the arrangement in Fig. 8, the induced volts per mile per conductor would be as given in equation (11), namely, 36.25, assuming the triangular arrangement of wires, with distance d between them. The arrangement shown in Fig. 8 is therefore slightly better from the point of view of inductive drop, notwithstanding that both sets of wires can be run on the same pole line with no greater spacing between wires than the minimum distance d determined by the voltage between phases. The fig- 374 ELECTRIC POWER TRANSMISSION ure 35.23 volts for the split three-phase system may be compared with 34.11 volts as given in equation (14) relating to the single- phase transmission with two circuits. It is clear that in either example, the drop in volts per conductor in the undivided circuit, with each conductor of sufficient section to carry the total current of 100 amp., would be 68.34, as given by equation (10). APPENDIX II SPECIFICATION FOR WOOD POLE TRANSMISSION LINE 1. General Description of Transmission Line. This transmis- sion line, which is 25 miles long, connects the water power gen- erating station at in the mining district of with the substation at the mines. The system will be three phase with a pressure of 22,000 volts between wires supported on wood poles. The conductors will be No. 2/0 stranded alumi- num. The average span will be 150 ft., and the separation between wires will be 3 ft., the three conductors being arranged in the form of an equilateral triangle with one conductor at the top and the remaining two conductors below, at the ends of a wooden cross-arm, all as shown on drawing No Where long spans are necessary, a double-pole arrangement, as shown on drawing No , will be adopted. Particulars relating to special precautions and methods of procedure in the case of exceptionally long spans, will be found in clause (8) under the heading "Spans." There will be no telephone wires supported on the transmission- line poles. There will be no grounded guard wire above the conductors, but galvanized iron lightning rods, as shown on drawing No , will be fitted to every third pole on the average. Further par- ticulars relating to protection against lightning are given in clause (7) under heading "Grounding." For particulars of sags and tensions, refer to clause (12) under heading "Stringing of Wires." 2. Clearing. The width of the right-of-way shall be 100 ft., and all lumber, brush and other growth of every description must be cut and cleared so that, in no portion of the right-of- way, shall the tops of tree stumps, undergrowth or bush be higher than 18 inches above ground level. At all points where a space of 50 ft. on each side of the pole line is insufficient to prevent possible damage to wires by falling trees, the normal width of the clearing must be exceeded. All useful material shall be separated and suitably stacked at a safe distance from waste material piled for burning. 375 376 ELECTRIC POWER TRANSMISSION All tops, limbs, brush and other waste shall be burned, great care being taken to prevent spread of fire beyond the limits of clearing. Suitable fire-fighting appliances shall be kept at hand while burning is proceeding. 3. Poles. Cedar poles shall be used when obtainable; but, owing to difficulties of transport, it is proposed to make use of the poles (mainly pine) obtained in the neighborhood of the trans- mission line while clearing the right-of-way. Dimensions. The greater number of the poles required will be 35 ft. long: they shall be sawn square at both ends. These poles shall measure not less than 24 in. in circumference at the top under bark, and not less than 38 in. under bark 6 ft. from butt. The approximate number required will be 840. In addition to these, about 100 poles 45 ft. long will be required, and these shall measure not less than 24 in. in circumference at top and 42 in. 6 ft. from butt. Quality. All poles to be cut of best quality live green timber, well proportioned from butt to top and well seasoned; the bark to be peeled, and all knots and limbs closely trimmed. The poles shall be reasonably straight, and no poles having short crooks or a reverse curve will be accepted. The amount of "sweep" measured between six-foot mark and top of pole shall not exceed 8 in. in the 35-ft. poles, or 11 in. in the 45-ft. poles. Twisted Poles. No poles having more than two complete twists in the total length, and no cracked poles will be accepted. Dead Poles. No dead poles or poles having dead streaks covering more than one-quarter of their surface will be accepted. Butt Rot. This must not exceed 10 per cent, of the cross-sec- tion of the pole, and the diameters of poles with butt rot or hollow hearts must be substantially greater than the corresponding diameters of sound poles. Poles with hollow hearts exceeding 8 in. in diameter will not be accepted. If average diameter of rot does not exceed 6 in., the butt measurement must be 2 in. greater than in the case of sound poles. If the average diameter of rot is 7 in., the butt measurement must be 4 in. greater. Miscellaneous Defects. Poles with sap rot, woodpeckers' holes, plugged holes, also poles that have been attacked by ants, worms, or grubs, are liable to be rejected as unsuitable. The treatment of all poles before erection shall be as follows: The gains shall be sawn square with the axis of the pole and in such a position that, when erected, the curvature of the pole WOOD POLE TRANSMISSION LINES 377 (if any) shall be in the direction of the line. The position of the gains is indicated on the accompanying drawing No showing the standard pole construction. The gains shall be not less than % in. and not more than % in. deep; they shall be accurately cut so that the cross-arms will have a driving fit, and the holes for the %-in. bolts securing cross-arm to pole shall be bored after the cross-arm has been fitted in position. These holes, together with all other necessary holes, as indicated on drawings, shall be bored clean and true without splintering. The holes for lag screws securing braces to poles shall be bored after braces have been fitted to cross-arms; they must be small enough in diameter to ensure that the threads of the lag-screw shall engage properly in the wood. The butts of all poles, together with the gains and tops, shall be treated with two coats of coal-tar-creosote oil, heated to about 220 F. and applied with a brush. At least 24 hours must be allowed to elapse between applications. The painting of the butts shall be carried at least 18 inches above ground level. 4. Cross-arms. The cross-arms shall be of yellow birch, Oregon fir, or long-leaf yellow pine, well seasoned, close grained, and free from knots or sap wood*. They must be dressed on all sides. They must measure 4J^ in. deep by 3^ in. wide, and be bored, as indicated on drawing No , with templet, true and symmetrical: the holes to be bored clean and without splintering. After having been bored, the cross-arms shall be painted with two coats of good asphaltum paint. In cases where double cross-arms are required, it will be necessary to bore the standard cross-arms with additional holes for the %-in. spacing bolts, the position of which is shown on the pole drawings previously referred to. 5. Grading. An effort should be made to maintain as far as practicable an even grade. By carefully choosing the location of each pole so as to avoid the highest points and greatest depres- sions when passing over uneven ground, it may be possible to avoid the use of poles differing in length to any great extent. Should it be necessary to shorten a pole, this must be done by sawing a piece off the butt end; but unless this is done before the treatment with preservative liquid, the butt must receive a further treatment with the creosote oil before erection of the pole. In some cases where the ground is favorable, the shortening of poles may be avoided by digging the hole deeper than would 378 ELECTRIC POWER TRANSMISSION otherwise be necessary. When using shortened poles, and when passing over uneven ground, it is important to bear in mind that under no condition shall the bottom conductors hang lower than 18 ft. above the ground, and when crossing tote roads or public footpaths, the minimum distance between wire and ground shall be 21 ft. 6. Pole Setting. Where poles are set in good solid ground, the depth of holes shall be as follows: 35-ft. poles on straight runs 5% ft. 45-ft. poles on straight runs 6 ft. 35-ft. poles at corners or where stresses are excessive . . 6 ft. 45-ft. poles at corners or where stresses are excessive . . 6% ft. If the ground is soft, the depth of setting shall be 6 in. greater than when setting in solid ground. If the soil is very soft, but not such as would be described as swampy, one or more transverse logs may be bolted to the butt of the pole in order to obtain addi- tional bearing area. When erected in solid rock, the depth of hole shall not be less than 3% ft. In loose or sandy soil, the sand barrel or its equivalent should be used. This must be filled with a firm soil which may contain stone or rock. In swampy ground the base of the pole must be provided with an arrangement of transverse timbers securely braced to the pole, in addition to which the hole shall, if necessary, be lined with sheet piling and filled with good soil which may contain stones or rock. As an alternative, a stone- or rock-filled crib may be built round the butt of the pole above ground level. In some cases concrete may be used with advantage in the pole foundation, but it will generally be found that the use of concrete can be avoided. Poles must not be set along the edge of cuts or embankments or where the soil is liable to be washed out, unless special precau- tions are taken to ensure durable foundations. When setting the poles in good ground, the holes shall be dug of ample size to allow of easy entrance of the butts, and the size at bottom must be large enough to admit of the proper use of tampers. When back-filling holes, there should be not less than three tampers to one shoveller, in order to ensure that the dirt shall be packed tight. In no case must the earth be thrown in to a greater depth than 6 inches without being tamped hard before WOOD POLE TRANSMISSION LINES 379 the next layer is thrown in. The proper filling of holes is a matter of great importance. When the filling is properly done, it should not be necessary to remove any excess soil ; this should be packed firmly around the pole, the object being to raise the level of the ground near the pole and so cause water to drain from, rather than toward, the butt. When setting poles on a straight run, the lining up should be done with a transit, and the poles placed with cross-arms truly at right angles to the direction of the line. Where the direction of the line alters, the poles at the angles must be set so that the cross- arm halves the angle. If the deviation exceeds 5 degrees, the corner poles shall be provided with double cross-arms and fixtures. When possible, the cross-arms, braces, and other fixtures (but not the insulators) should be mounted on the poles before erection. 7. Grounding. The proper grounding of lightning rods on the pole line is a matter of importance. Judgment must be used in determining when and how to ground the poles; but either of the following alternative methods will be considered satisfactory, provided the soil is reasonably moist: (1) A piece of galvanized iron pipe 1^ in. in diameter and 8 or 9 ft. long shall be buried in the hole alongside the butt or driven into soft soil, the ground wire being attached thereto in such a manner as to ensure a good and enduring electrical contact. (2) The ground wire, consisting of %$ in. galvanized stranded steel cable, after being carried straight down the side of the pole and secured with cleats, shall be wound spirally around the butt and carried right down to the bottom of the pole. Not more than 15 ft. of wire should be buried in the ground. It is of little use to ground a pole in solid rock, but where a pole is set in rock, it may be found that the ground wire can be carried down the face of the rock, or in a crevice, to a point where a good ground can be obtained. Where grass is growing, the soil will usually contain sufficient moisture to afford a reasonably good ground. When the ground wire does not enter the ground alongside the pole, sudden bends or turns should be avoided in the wire connecting the lightning rod with ground plate or pipe. It is not intended to provide all poles with lightning rods; but, except when the soil is clearly unsuitable for a ground con- nection, the poles in the positions described below shall be grounded : 380 ELECTRIC POWER TRANSMISSION Both poles supporting extra long spans requiring the double pole arrangement as shown on drawing No previously referred to. The poles on each side of railway crossings. All guyed poles. The six poles nearest to generating station. The six poles nearest to substation. In addition to the above-mentioned poles, one pole out of every three poles shall be grounded. It is not necessary that every third pole be grounded: judgment must be used in deter- mining the location of the poles to be grounded. As a general rule, it is more important to ground poles on heights and in exposed positions than those on the lower ground; but, on the other hand, it is of little advantage to ground where the soil is dry or otherwise unsuitable. In exposed positions it may be advisable to ground two or more consecutive poles, while in unex- posed positions four or five consecutive poles may be left without lightning rods. 8. Spans. The standard length of span shall be 150 ft. Shorter spans must be used at angles and on curves, as mentioned in clause 9. If the span exceeds 1 70 ft. the poles must be specially selected for strength. No span greater than 190 ft. shall be carried on single poles. For longer spans, the double-pole arrangement as shown on drawing No previously referred to, shall be adopted, with a horizontal spacing of 5 ft. between wires for spans up to 600 ft. ; but spans exceeding 500 ft. shall be avoided if possible. Railroad Crossings. (a) The span where line crosses railroad shall be kept as short as possible ; but hi no case must a pole be placed a smaller distance than 12 ft. from the rail, except in the case of sidings, where the distance may be reduced to 6 ft. At loading sidings sufficient space must be allowed for a driveway between rail and pole. When possible the distance between rail and pole should not be less than the height of the pole, but if this spacing requires a span greater than 120 ft., it will be prefer- able to place the pole nearer to the rail provided the ground is suitable. If it is necessary to cross the railroad with a span greater than 150 ft., the double-pole arrangement as used for extra long spans, and as shown on drawing No shall be adopted. (6) In all cases the cross-arms and insulators shall be doubled on the poles nearest the rail. WOOD POLE TRANSMISSION LINES 381 (c) The poles at railroad crossings must be set not less than 6 ft. in the ground (4 ft. in rock). (d) If the crossing is at a spot where grass or other fires might cause injury to the poles, these shall be provided with a casing of concrete at least 2 in. thick, to a height of 5 ft. above ground level. (e) The clearance between rail and high-tension conductor shall not be less than 30 ft., and the poles should be specially selected for strength and straightness. (/) When crossing over telephone wires, the clearance shall be not less than 10 ft. (g) The poles at railway crossings must be securely guyed, whether or not there is a bend in the line. If a departure from the straight run is necessary, special attention should be paid to the method of guying. (h) The poles on each side of the rail shall be provided with lightning rods, and well grounded. Bent iron lightning guards, as shown on drawing No shall be fixed at each end of cross- arm and connected to the ground wire; these will also serve the purpose of hook guards, to engage the conductor if it should become detached from the insulator. If the nature of the soil is quite unsuitable for the purpose of grounding, the lightning rod may be omitted; but if the pole is not grounded, two strain insulators must be placed in each guy wire securing poles nearest to rail; the upper of these insulators being not less than 6 ft. distant from the lowest high-tension conductor, and the second insulator being not less than 8 ft. above ground level. (i) Special attention shall be paid to the tying of the con- ductors to the double insulators on the poles at each side of the rail. As a protection against damage by arcs over insulators, the serving of No. 2 aluminum tie wire shall be carried far enough to ensure that the conductor is protected by the serving or tie to a distance of not less than 12 in. from the center of insulator. ( j) In addition to the pole number, the poles on each side of the crossing shall bear a label with the Company's name and the voltage (22,000 volts) painted thereon in easily distinguish- able characters. 9. Angles and Curves. Whenever there is a change in the direction of the line, a sufficient number of poles must be provided to prevent the angle of deviation on any one pole exceeding 15 degrees. If the deflection from the straight run does not exceed 382 ELECTRIC POWER TRANSMISSION 5 degrees it is not necessary to use a pole with double fixtures. When the deflection exceeds 5 degrees, poles with double fixtures shall be used, and these must be side guyed. When the "pull" at corner pole exceeds 2 ft. the span on each side of pole shall be less than 150 ft.; the reduction in the length of span being at the rate of about 2}/ ft. per foot of "pull," all as indicated in the table accompanying Fig. 1. Should it be necessary to turn the line at a point where space is limited, through an angle greater FIG. 1. Limit of length of spans on each side of angle pole (standard span = 150 ft.). "Pull" D feet Deflection a degrees Short span, S, not to exceed: feet Remarks 2 2-18' 145 | 3 3-26' 143 Double fixtures not nec- 4 4-36' 140 essary. 5 5-44' 138 6 6-52' 135 7 8-00' 133 8 9 9-10' 10-20' 130 128 Use double fixtures on 10 11 ll-28' 12-38' 125 123 poles. Side-guy. 12 13-47' 120 13 14-56' 118 than 15 degrees, two or more poles with double fixtures may be set close together, each pole being side guyed, or securely braced. In all cases where there is a departure from the straight line, the poles must be set so that the cross-arms will bisect the angle. 10. Guying. The material to be used throughout for guys is Ke-in. galvanized seven-strand steel cable. Where the wire is wrapped around the pole, a protecting strip of No. 24 galvanized sheet iron shall be put under the wire. The wire shall make two complete turns about the pole. WOOD POLE TRANSMISSION LINES 383 The anchoring shall generally be done by burying an anchor log from 4 ft. to 6 ft. long, and bolting thereto a %-in. guy rod. Other methods may have to be adopted to suit the varying nature of the ground, but in all cases it is important to ensure a good hold and to see that the guy rod is in line with the guy wire. The angle of the guy wire when anchored in the ground shall be approximately 45 degrees where circumstances permit. No strain insulators shall be used on guy wires, except as called for at railway crossings; but all guyed poles shall be provided with lightning rod and be well grounded. It is not intended that work be done on live wires on guyed or other grounded poles. As a general rule all poles shall be guyed before the conductors are strung. Poles must be guyed at all points as mentioned below: (a) At angles exceeding 5 degrees. (6) Where the line goes up a 15 per cent, or steeper grade (head guys every fifth or sixth pole, or only at top of hill on short lengths). (c) On hillsides where the footing may be good, but where there is danger of slipping stones or soil producing side pressures on the pole (side guy). (d) At each end of exceptionally long spans, where double poles are used. (e) All poles with double fixtures. 11. Insulators. The line insulators will be supplied by Messrs. They will be of the pin type, the pins having porcelain bases with wood thimbles and ^-in. galvanized iron bolts for fixing to cross-arms. The pole-top insulators will be supported on malleable-iron pole-top pins, and the separable thimbles of these pole-top pins will be cemented into the insu- lators at the makers' works. The insulators shall be mounted on the cross-arms after the poles have been erected. The pole- top insulator pins may be bolted in position before erection of pole. 12. Stringing of Wires. No. 2/0 seven-strand bare aluminum cable will be used throughout. Care must be used in handling the conductors, to guard against cuts or scratches or kinks. The conductor must not be drawn over rough or rocky ground where it is liable to be injured by stones, etc. It is important that the cables be pulled up to the proper tension so that the sag will be in accordance with the particulars 384 ELECTRIC POWER TRANSMISSION given on the curves Figs. 2 and 3. These curves give not only the correct sag at center of span, but also the required tension in the cable at the time of stringing. The curves are calculated for wires subject only to their own weight and hanging in still air. In the case of extra long spans, and where the grade is not constant, it will generally be found more convenient and quicker to adjust the tension by means of a spring dynamometer than by measuring the sag. The cables must not be pulled around insulator pins on angle poles. 100 90 80 r g 60 1 50 I 40 ft 30 ^20 1 10 I -10 -20 -80 \ \ V \ ^ Tension Temperature Curves For use in Stringing No.2/0 Stranded Aluminium Conductors (Calculated for Maximum Stress=13000 Ibs. per sq. in. with wind velocity 47 miles per hour, combined with M"ice coating, at-20F. ) ^ ^x \ \^ \^ X. \ X X X \ N X X. X x ^X x X X ^ X x X X X ^ X X X ^ \ X \^_ X v^ v X X ^ ^? ^ **/*/ Sft ^ x. X \ >. -s N, \^ ^ ^ SJ 1 s ^s X ^ \ "X \ X, 200 300' 400 500 600 700 800> 900 -10( " Tension in Lbs.as Indicated .by Dynamometer FIG. 2. Chart giving tension at which wires should be strung. 1100 The tie wire shall be No. 2 B. & S. solid, soft aluminum wire. The tie on straight runs shall be of the type known as the armor top, with the conductor in the groove on top of insulator. At corners, the tie shall be of the type known as the armored Western Union, with the conductor carried around the insulator in the side groove. The tie shall be a modification of the type used by the Niagara, Lockport and Ontario Power Co., between Niagara Falls and Buffalo. The serving of No. 2 tie wire on the conductor is for the purpose of preserving the latter from abrasion and from damage due to possible electric discharges over the insulator. The use of pliers should be avoided in making the ties, except for WOOD POLE TRANSMISSION LINES 385 the final clinching, when they must be used with care to avoid cutting or otherwise injuring the conductor. When joints are required in the conductors, they shall be made with Maclntyre tubes which shall be given two twists with the splicing clamps provided for the purpose. 100 90 80 ITO leo 50 I- 830 p .10 1. -10 -20 -30 P / 4 4 v> N g r^ jA V / S/ Y *> ^ *rf ? / f . / / ' / S / / / / / / / / s / / / / / jf / / / / 7 / / / ] / / / / / / 1 / Sag Temperature Curves For use in Stringing No.2/0 Stranded Aluminium Conductors (Calculated for Maximum Stress=13000 Ibs. per sq. in. with wind velocity 47 miles per hour, combined with H"ice coating, at-20'F.) / 1 / 1 4 8 10121416182022242628303234363840 Sag at Center of Span (Inches) FIG. 3. Chart giving sag in wires when correctly strung. 13. Locating and Numbering Poles. All poles shall bear a distinguishing number in clear bold figures about 10 ft. above ground level. These numbers will correspond with the numbers on the plans which will be prepared as soon as possible after the poles have been erected in position. ' The plans will be drawn to a scale sufficiently large to show the location of each pole. 25 APPENDIX III SPECIFICATIONS FOR STEEL TOWER TRANS- MISSION LINE These specifications, which are preliminary specifications sub- ject to revision in minor details after bids for the various materials have been received and considered, cover the construction of an overhead transmission line connecting the generating station at with distributing station at ^ a distance of approximately 60 miles as measured along the right- of-way of the transmission line. Methods of construction are not dealt with in detail, because the work hi the field will be in the hands of a competent and ex- perienced construction engineer who will be allowed considerable latitude in regard to the actual handling of materials and in de- ciding upon the best methods to be adopted in the erection of towers, stringing of conductors, and other details of practical line construction. General Description of Line. Two three-phase circuits of No. 2/0 equivalent copper cable will be run in parallel on one set of steel towers spaced approximately eleven to the mile. The towers will be of the semi-flexible type, with rigid strain towers at intervals of about a mile, or more frequently where corners or extra long spans render then: use necessary. A J^Q-in. gal- vanized Siemens-Martin steel-strand cable will be carried the full length of the line and be firmly secured to the top of each tower. The pressure between conductors will be 80,000 volts, and the suspension type of insulator will be used throughout. Details of entering bushings and methods of connecting light- ning arresters at generating and receiving stations are not dealt with in these specifications as they come under another contract. The proposed line has been staked out by the stadia survey party, and the right-of-way secured where necessary. Stakes have been driven to indicate proposed location of towers, but these positions are subject to modification. 386 STEEL TOWER TRANSMISSION LINES 387 The line passes through country that is for the greater part uncultivated; the ground is undulating and in some parts wooded. A considerable amount of clearing has yet to be done. Roads are bad; but the transmission line is within 2 to 3 miles of the railway at all points. Duties of Engineer in Charge of Construction. Before the work of construction is begun, the construction engineer will go over the line as staked out by the preliminary survey party and as shown on plan No herewith. He will take with him an engineer equipped with a light transit, and an assistant capable of acting as axeman or rodman as circumstances may require. The construction engineer will decide in the field the position of each tower, making changes in the preliminary plan in the matter of tower locations and even to a small extent in the route to be followed, if in his opinion such changes will result in a better and more economical line. Hub-stakes shall be driven to mark the center-point of each tower, and a second stake shall be driven about 12 ft. ahead or in the rear of the hub-stake in the direction of the line. This is for reference when setting the anchor stubs. The construction engineer must check clearances between conductor and ground, and on long spans, especially if there is doubt as to position and amount of minimum clearance, he should take the necessary particulars to allow of the matter being settled in the office. After agreeing and checking the alterations to plan in the office, the construction engineer will assist in making out the shipping schedules for delivery of materials at the most suitable points. In regard to the work of erection proper, the construction en- gineer will attend to all details of organization of the parties in the field, and will study the best means of distribution of materials along the line ; all with the view of avoiding unnecessary expendi- ture, and of carrying out the work expeditiously and in a work- man-like manner. Such details as the actual methods to be adopted in the erection of towers and stringing of wires will be decided upon after discussion with the chief engineer, and after due weight has been given to manufacturers' suggestions. Clearing. On those parts of the line on which clearing is re- quired, it is proposed that this work be done immediately after the line has been finally staked out. This clearing will extend 60 ft. on each side of the center line of the right-of-way, and it will be carried out under a separate contract. 388 ELECTRIC POWER TRANSMISSION Towers. Two standard types of steel tower will be used : these will be referred to as the strain type and flexible type respect- ively. Copy of the specification on which bids will be obtained from manufacturers is attached hereto. Foundations for Towers. The use of concrete is to be avoided, but in marsh land or loose soil concrete footings may be necessary. The decision as to where concrete is to be used will rest largely with the engineer in charge of construction in the field. When the tower stands on solid rock which may occur in a few instances the standard footings will not be used; but a special wedge bolt, shaped at the top to take the standard tower, will be grouted in with sulphur or other approved cement. In levelling up on rock foundations, it may sometimes be cheaper to build up one or two piers of concrete, securely tied down to the rock, rather than level off the rock on the high side. In selecting sites for towers, the construction engineer shall pay attention to the matter of foundations, and endeavor to secure sites where the foundations are good. Hillsides are to be avoided, especially where the soil is liable to crumble or slide. The matter of grading should also be considered when finally selecting sites: much may be accomplished in the judicious selec- tion of tower sites by slightly adjusting the length of span to obtain sites which will tend to equalize the grade. To facilitate the work of erection, a wooden digging templet will be provided, together with a rigid but light-weight angle steel templet to ensure the correct placing of the anchor stubs; the latter being bolted to the templet before the work of back- filling the holes is commenced. The second stake which, as previously mentioned, will be driven truly in line with the hub stake, will be used for the correct setting of these templets. The steel templet must be carefully leveled up in order that the center line of the tower shall be vertical. Grounding. In cases where the iron work of the foundations is completely encased in concrete, the tower shall be well grounded by means of a 10-ft. length of 1-in. galvanized-iron pipe driven or buried in the ground, and electrically connected to one of the tower legs. When the tower stands on rock, an effort should be made to obtain a good ground by carrying a length of the galvan- ized guy wire from the tower leg to a rod driven in damp soil at a short distance from the tower if a suitable spot can be found. STEEL TOWER TRANSMISSION LINES 389 Guying. Where guy wire is required, the Ke- m - Siemens- Martin steel ground wire shall be used. When the distance between strain towers exceeds % mile, one flexible tower situated about midway between the strain towers shall be head-guyed in both directions. Flexible towers used at corners where the deviation lies between 5 and 8 degrees, and the approach spans are of normal length, shall be guyed with two guy wires so placed as to take the corner strain and resist overturning of the tower owing to the resultant pull of the wires. Angles. The semi-flexible support is designed for use on straight runs only; but if the deviation from the straight line does not exceed 5 degrees, these intermediate supports may be equipped with strain insulators and used at corners. For angles greater than 5 degrees, but not exceeding a limit of 8 degrees, these supports may be used with two guy wires to take the transverse stress due to the resultant pull of the wires, if the approach spans are reduced to 240 ft., an 8-degree curve may be turned on a semi-flexible structure without guy wires. Strain towers shall be used for turning corners up to 30 degrees ; but when the total deviation exceeds this amount, two towers must be used. Erection of Towers. The actual organization of the various crews for distributing material, setting anchor legs, assembling and erecting the towers, will be left to the engineer in charge of construction, who will so conduct operations as to carry out the work efficiently at the lowest possible cost. 1 Insulators. These will be of the suspension type to comply with the requirements detailed in the insulator specification of which copy is attached hereto. Conductors. The conductors shall be 19 strand hard-drawn copper cables equivalent in section to 00 B. & S. gauge. The tensile strength of the finished cable shall not be less than 90 per cent, of the strength of the individual wires forming the cable; and these shall satisfy the strength requirements of the standard specification drawn up by the American Society for Testing Materials. 1 Mr. R. A. Lundquist's book on "Transmission Line Construction" is of special interest to the engineer in charge of construction. Excellent articles describing practical methods of construction also appear from time to time in the technical press. The article by Mr. A. B. Cudebec on "Steel Tower Transmission Line Construction" in the Electrical World of July 17, 1915, contains much valuable information. 390 ELECTRIC POWER TRANSMISSION The electrical conductivity shall not be less than 97 per cent, by Matthiessen's standard. The total weight of copper conductor required is estimated at 800,000 Ib. It shall be delivered on drums or reels each contain- ing 1 mile of cable. Joints in Conductors. The splices shall be made with copper sleeves of the "Maclntyre" or similar approved type. The finished joint shall consist of 3 turns. The tools provided for the purpose shall be used in making the joints. Spans and Wire Stringing. The actual method of stringing the wires will be left to the judgment and experience of the con- struction engineer. It is, however, suggested that three conduc- tors be drawn up at a time, using the arrangement of sheave blocks known as an "equalizer." The average span shall be approximately 480 ft. This may be increased to a limit of 500 ft. between flexible supports, and to a limit of 1200 ft. between two strain towers without intermediate supports. It is thought that three or four points on the line may advantageously be spanned between two strain towers placed from 1000 to 1200 ft. apart. In the case of abnormally long spans, it is important to see that the contour of the ground is such as to allow of maximum sag while maintaining the specified mini- mum clearance between H. T. conductors and ground. The clearance between lowest wire and ground shall in no case be less than 28 ft. The charts Nos and give all necessary particu- lars for the stringing of conductors and guard wire at various temperatures. The guard wire connecting the tops of all towers shall be strung and securely clamped to the steel structure before the conductors are drawn up. Dynamometers will be provided, and their use is recommended, especially when spans are unequal in length, and on extra long spans between two strain towers. If an equalizer is used, it is not necessary to insert a dynamometer in more than one leg. Special attention shall be paid to the drawing up of cables to the proper tension or sag. Too great a sag is almost as objectionable as too great a tension; but it must be remembered that where a dip occurs in the line of supports there is sometimes a possibility, in very cold weather, of the conductor being drawn up (by contraction) above the proper level of the lowest insulator. The construction engineer should watch for this possibility with a view to guarding against it. STEEL TOWER TRANSMISSION LINES 391 When the conductors have been drawn up and transferred from snatch block to insulator clamp, it is important to see that the suspension insulator hangs truly vertical before finally tight- ening up the clamp. There will be no transpositions on the H. T. conductors. The telephone wires are run on a separate set of wood poles and they will be transposed at every support. SPECIFICATION FOR STEEL TOWERS These towers are for use on an 80,000-volt, three-phase trans- mission line using insulators of the suspension type. It is proposed to use two standard types of towers only; these will be referred to as the rigid or strain towers, and the flexible towers. The strain towers shall be designed with four corner legs and square bases, generally as indicated on plan No herewith. An effort will be made to avoid the use of special structures, and where extra long spans have to be carried, two standard strain towers may be placed close together. In one or two places it may be necessary to use extra high towers, and it is proposed to use the standard tower mounted' on a special base, generally as shown on plan No , designed to raise the tower 18 ft., or such other amount approximating to this dimension as may best suit manufacturers' designs. The intermediate or flexible type of support will be of the " A "- frame design, generally as shown on plan No Preference will be given to a design consisting of few parts, provided this will not add appreciably to the cost of transporting the towers over rough roads to the point of erection. The parts of all towers shall be galvanized when ready for assembling; but, in the case of the flexible type of structure, an alternative offer for painted steel work will be considered, provided the number of parts is small and the section of metal reasonably large. The plans referred to, which accompany this specification, give all necessary leading dimensions; but the cross-section of the vari- ous members and the details of design are left to the manufac- turer, who is also at liberty to submit alternative proposals. In no case must the distance between conductors be less than 8^ ft. or the height above ground of the point of attachment of insulators on lowest cross-arm less than 40 ft. The sections of structural steel used for the main corner members of the strain 392 ELECTRIC POWER TRANSMISSION towers or for the main members of the flexible towers shall not be less than ^ in. thick, and no metal less than %g in. thick shall be used in the construction of these towers. Number of Towers Required. Offers shall be based on the following quantities, which are subject to slight modification. Flexible towers (plan No ) 592 Rigid towers (plan No ) 65 Extension bases (plan No ) 4 Working and Test Loads for Towers. The normal length of span is 480 ft. and the total vertical load per tower, consisting of six conductors and one guard wire together with estimated possible ice loading and the weight of the six insulators, is 3100 lb.; but the spans will in many cases exceed the aver age length. The maximum total overturning pressure in a direction at right angles to the line, due to wind blowing across the wires, is esti- mated at 3300 lb. ; this may be considered as distributed equally between the points of attachment of the seven wires. The manu- facturer should estimate the pressure of wind on the tower structure itself by allowing a maximum pressure of 13 lb. per square foot of tower surface. . A factor of safety of 2^ shall be used in making stress calculations. One tower of each type shall be tested in the presence of pur- chaser's representative, and must withstand without exceeding the elastic limit of the steel, or suffering appreciable permanent deformation, the following test loads applied at the points indi- cated on the plans previously referred to. These tests are to be made with the tower erected on its own foundations in such a manner as to reproduce as nearly as possible the conditions under which it will ultimately be erected. Strain Tower Test Loads. (1) A breast pull of 15,000 lb. applied in the direction of the line at the point of attachment of the middle cross-arm. (2) A vertical load of 1000 lb. applied at the end of any cross- arm. (3) A torsional load of 3500 lb. applied in a direction parallel to the line at the end of any cross-arm. Flexible Tower Test Loads. (1) A transverse pull of 4500 lb. applied in a direction at right angles to the line at the point of attachment of the middle cross-arm. (2) A vertical load of 800 lb. applied at the end of any cross- arm. STEEL TOWER TRANSMISSION LINES 393 Metal steps shall be provided on all towers within 8 ft. of ground level for the use of linemen. It is requested that manufacturers tendering for steel towers call attention to any features of the particular design proposed which may tend to reduce cost of transport and erection on site, as these are matters which will receive consideration when placing the contract. Galvanizing Test. The purchaser reserves the right to reject all towers of which the galvanizing is not of the best quality. Tests will be made before erection as follows: Samples of steel work will be immersed in a solution of sul- phate of copper (specific gravity about 1.185) maintained at a temperature of 60 to 70 F. After remaining in the solution 1 minute, the sample will be removed, thoroughly washed in water, and wiped dry. This process will be repeated four times, after which there must be no appearance of red spots indicating copper deposit. SPECIFICATION FOR PORCELAIN LINE INSULATORS Number of Insulators Required. The approximate quantities required, as based on preliminary estimates are: Suspension type 4000 Strain type 880 Climatic Conditions. The transmission line on which the insu- lators will be used is located in the district, where severe thunder storms and heavy rain may be expected during the summer months, and where sleet storms and low tempera- tures are prevalent in the winter. Working Voltage. The transmission is three-phase off delta connected transformers, at a frequency of 60 and a maximum working pressure of 84,000 volts between wires. Design of Insulators. The design of the suspension type and strain insulators is left to the manufacturer, who must submit dimensioned drawings or samples with his offer. The units making up the strain insulators need not necessarily differ in de- sign from the units of the suspension insulators, provided the latter are capable of withstanding the mechanical tests required for the strain insulators. The towers have been designed on the assumption that the weight of one complete string of unit insula- 394 ELECTRIC POWER TRANSMISSION tors will not exceed 60 Ib. and that the distance between point of suspension and conductor will not exceed 36 in. These limits should not be exceeded. It is preferred that the number of units in the complete string be not less than three nor more than five. Metal Parts. All metal parts subject to rust and corrosion, such as malleable iron castings and steel forgings, shall be heavity galvanized and capable of withstanding the usual tests. Glaze. The surfaces of the porcelain not in contact with the cement shall be uniformly coated with a brown glaze, free from grit. Cement. Pure Portland cement only shall be used in as- sembling the parts of the unit insulator. Mechanical Tests. An inspection will be made of all insulators with the object of rejecting those containing open cracks in glaze or porcelain. One complete suspension insulator, selected at random, and consisting of the requisite number of units, shall withstand a load of 5000 Ib. without rupture or sign of yielding in any part. At least three units of which the strain insulators are built up shall be tested to the breaking limit, and must withstand an ulti- mate load of not less than 12,000 Ib. Electrical Tests. Three or four complete insulator strings, both suspension type and strain type, shall withstand without flash over a " wet " test of 200,000 volts. In all cases the electrical stress shall be applied for 1 minute, and the spray shall be directed upon the insulator at an angle of 45 degrees under a pressure of 40 Ib. per square inch at the nozzles, the precipitation being at the rate of 1 in. in 5 minutes. The suspension insulators shall be hung vertically, and the strain insulators horizontally. The connection of the test wires shall be so made as to re- produce as nearly as possible the working conditions. The manufacturer shall satisfy himself by his standard factory tests that each unit is sound mechanically and electrically. The "dry" flashover of the complete string of insulator units shall not be less than 240,000 volts; but this test need not be made in the presence of the purchaser's representative. The transformer used for the electrical tests must be capable of a reasonably large k.v.a. output; the e.m.f. wave shall be as nearly as possible sinusoidal, and the frequency shall be within the limits of 25 and 60 cycles. STEEL TOWER TRANSMISSION LINES 395 Packing of Insulators. It is desirable that the parts for one complete insulator, or at most for two insulators, be packed complete in a separate barrel or crate, and that the contents be clearly described on attached label. Wire Clamps. A suggested clamp for use with suspension insulators is shown on drawing No herewith; and drawing No shows a proposed strain insulator clamp. Makers are asked to submit samples or drawings of their standard types, preferably of the general design indicated by the above-mentioned drawings. The conductor to be carried is an equivalent No. 2-0 gauge (B. & S.) stranded copper cable; the groove for the wire should be slightly curved and flared at the ends. INDEX Aluminum cell arresters, 187 conductors. (See Wire.) compared with other mate- rials, 48, 69 Anchors for guy wires, 383 Angles and curves on transmission lines, 381, 389 Arcing-ground suppressor, 194 rings and horns, 177 Arresters. (See Lightning arrest- ers.) Asphalt troughing for underground cables, 208 Auxiliary steam-driven plant, 43 B Bi-metallic conductors, 69, 72 "Boosters" for voltage control, 113 Brush treatment of pole butts, 311 Bushings, condenser type, 150 entering, 143 insulating, design of, 146 Butts, pole, preservative treatment of, 309 reinforcing decayed, 312 Cables. (Refer also to Wire.) overhead, stranded compared with solid wires, 263 with hemp core, 73 with steel core, 73 underground, 6, 9, 38, 199 advantages for D. C. trans- mission, 244, 200 as protection against surges, 191 capacity of, 98, 212, 217 Cables, underground, construction of, 202 cost of, 199, 208 design of, 212, 221 grading, 216 joints in, 228 losses in, 223 methods of laying, 206 reactance of, 214, 217 submarine, 200 temperature rise of, 225 terminals, 211 voltage limitations, 201 Capacity, 130 current, 29, 31, 99, 217, 223 line losses due to, 112 distributed, 101 effect of, on regulation, 28 in line insulators, 135 in terms of inductance, 100 of suspension insulators, 135 of three-phase lines, 30, 32, 97 of transmission lines, 28, 96 of underground cables, 98, 212, 217 pressure rise due to, 29, 109 Catenary curve for sag calculations, 249 Charging current. (See Capacity current.) Choke coils, 192 Clearance between conductors and pole, 158 between entering wires and building, 144 between overhead wires, 157 Clearing ground, 375, 387 Coefficient of linear expansion, 75 of self-induction, 88 Compression members of towers, stresses in, 337 wood pole in, 323 397 INDEX Concrete for tower foundations, 344 poles. (See Poles, concrete.) Condensers for lightning protection, 189 Condenser type of bushing, 150 Conductivity of conductor materials, 75 Conductors. (See Cables: Wires.) physical constants of, 74 Conduits for underground cables, 207 Constant current system. (See Thury system.) pressure transmission, 119 Continuity of service, 3, 7, 42 Continuous currents, transmission by. (See Thury system.) Copper-clad wire, 69, 72 conductors, 70 Corona, 151 as "safety valve," 156 losses due to, 152, 154 voltage formulas, 154 Costs. (Refer to subject or item.) Creosote oil for wood preservation, 311 Critical temperature (sag calcula- tions), 284, 286 voltage of corona formation, 153 Cross arms, 139, 377 Current density in underground cables, 225 total, in line with appreciable capacity, 110 Curves and angles on transmission lines, 381, 389 Deflection. (See Sag; Poles; Tow- ers.) Depreciation, 50, 53, 55, 59, 61 Dielectric circuit (fundamentals), 130 constant, 99, 130, 213 flux, 130 Direct current transmission. (See Thury system.) Distance of transmission (economic limits), 36 Disturbances due to switching op- erations, 173 Ducts for underground cables, 207 Duplex wire (copper-clad steel), 69, 72 Duplication of transmission linefc, 3, 10, 42, 368, 373 E Earthing. (See Grounding.) Economic conductor section, 48, 55 considerations, general, 2, 5, 7, 10, 12, 13, 36, 60 ohmic pressure drop, 52, 65 voltage, 7, 53, 59, 66 Elastance, 147 Elastic limit (various materials), 75 modulus, 75 transmission line. (See Flexi- ble.) Electrolysis, 231 Electrostatic induction, 122 Energy stored in magnetic and elec- tric fields, 164 Erection of poles and towers, 378, 359, 389 Erection of wires. (See Wires, stringing of.) Estimates for complete overhead lines, 43 Factors of safety. (See Safety fac- tors.) Farad, 130 Faults on overhead transmissions, 125, 162 in underground cables, 228 Flash-over distances (insulators), 133 Flash-over voltage (suspension insu- lators), 138, 140 Flexible steel towers, 37, 331, 343, 349, 392 tower lines, 331 INDEX 399 Flux, dielectric, 130 Foundations. (See Pole; Tower.) Frequency in relation to telephone troubles, 125 Gains, 377 Galvanizing, test for, 393 versus painting for steel towers, 41 Garton-Daniels lightning arrester, 186 Glass insulators, 127 Grade, lines carried up steep, 261 "Grading" horn lightning arresters, 183 Grading transmission lines, 377 underground insulated cables, 216 Graphical statics applied to sag-ten- sion calculations, 249 Grounding, methods of, 178, 379, 388 neutral of three-phase trans- mission, 23, 120 Ground resistance, 238 Guard rings on insulators, 177 wires, 176, 196 Guying steel towers, 348 wood poles, 382 Inductance of three-phase lines, 33, 363 with any arrangement of con- ductors, 34, 361 Induction, electrostatic (telephone interference), 122 magnetic, 123. (Refer also to Inductance; Reactance.) Insulation, comparative, of A. C. and D. C. transmissions, 237 Insulator materials, 127 Insulators, cost of, 42 design of, 129, 132, 142, 393 deterioration of, 162 "electrose," 127 factors of safety, 159 flash-over voltage, 160 glass, 127 pin type, 131 porcelain, 127 rating of, 161 suspension type, 134 testing, 302 ties for, 302 weight of, 162 Interruptions to service, 3, 7, 42 " Intersheath " in insulated cables, 203 Iron (or steel) for overhead conduc- tors, 69, 71, 83, 300 H Hemp core cables, 73 Horn gap lightning arresters, 180 Howard asphalt troughing, 208 Ice and snow, effects of, on wires, 5, 265, 275 Impedance of power lines. (See Re- actance, Inductance.) "natural," 165, 173 Inadequacy, 61 Inductance of power lines, 14, 24, 79, 84, 88, 361 in terms of capacity, 100 Joints in cables, 228 in overhead conductors, 301, 390 Junction between overhead and un- derground conductors, 211 K Kelvin's law, 36, 49, 57, 66 Leads, entering, 143 Life of poles and towers. (Refer to subject or item.) 400 INDEX Lightning arresters, 175 aluminum cell, 187 condenser type, 189 Garton-Daniels, 186 horn gap, 180 low-equivalent, 183 multi-gap, 183 separation of, 192 water jet, 179 protection of overhead lines from, 163, 174, 195 rod, 175 Line drop, 53, 104, 108, 363 Loading of wires, usual assumptions, 270 Losses in transmission lines, 13, 20, 22, 56, 57, 64, 110 cost of, 50, 53, 57, 64, 66 underground cables, 223 M Magnetic induction (telephone inter- ference), 123. (Refer also to Inductance; Reactance.) Materials for conductors, 48, 69 Mershon diagram, 90 Modulus of elasticity, 75, 279 Mosciki condensers, 190 N Natural frequency, 167 impedance of line, 165 Parabola and catenary compared, 248 Patrolling transmission lines, 9 Permittance, 130 Physical constants of conductor materials, 74 Pin type insulators, 131 Pole butts, preservative treatment of, 309 reinforcing decayed, 312 foundations, 320 Pole lines, wood, for high pressures, 4, 48, 303 typical, 4, 306 specifications for, 375 Poles, concrete, 303, 324 cost of, 39, 325 factors of safety, 329 life of, 39, 324 strength and stiffness of, 327 weight of, 325 corner, load carried by, 322 steel, 4, 37, 301, 303, 329, 334 wood, 300, 303, 305, 376 "A" and "H" type, 37, 303, 307 cost of, 39, 41, 48 deflection (stiffness) of, 318 depth of holes for, 321 factors of safety, 315 guying, 382 insulating qualities of, 313 life of, 39, 304, 307 preservative treatment of, 309 setting (erecting), 378 spacing of, 3, 37, 321 strength of, 314 weight of, 313 Potential gradient, 131, 147, 149, 152 "Pot-heads," 211 Power factor, 19, 21, 23, 104, 120 control of, 115, 117 of several circuits in parallel, 120 of three-phase circuit, 34 of underground cables, 224 losses. (See Losses.) maximum on a single transmis- sion line, 10 stations, cost of, 60 total, transmitted by polyphase system, 21 by three-phase line, 33 Preliminary work; planning new lines, 5, 387 Preservative treatment of pole butts, 309 Pressure available at intermediate points on line, 94 INDEX 401 Pressure, barometric, at various alti- tudes, 155 control, 112, 116 drop, 53, 104, 108, 363 limits on overhead lines, 158 on underground cables, 201 rise due to capacity, 29, 109 rises on lines r arrying large cur- rents, 165, 197 surges due to switching opera- tions, 165, 173 Props or struts (wood), 323 Protection of lines against over- voltages. (See Lightning.) Quarter wave length line, 163, 169 R Radius of gyration (tower design), 338 Railroad crossings, 9, 271 Reactance, 72, 83, 85, 87. (Refer also to Inductance.) of underground cables, 214 Reactors, rotary, to control voltage, 117 Regulation, voltage, 24, 56, 89, 108 effect of capacity on, 28 Reserve steam driven plant at re- ceiving end of line, 43 Resistance in ground connections, 23, 181 insulation, of cables, 213 losses due to, 14 necessary to prevent oscilla- tions, 166 of earth as return conductor, 238 table (conductor materials), 75, 76, 85 to alternating currents. (See Skin effect.) Resistivity of cable insulation, 214 Resonance, 167 Roof outlets, 143 S Safety factors, line insulators, 159 poles, 315, 329 towers, 392 wires, 7, 301 Sag, aluminum and copper com- pared, 298 calculation of, 251 effect of wind, ice, and tempera- ture on, 281 with supports at different eleva- tions, 257, 260, 290 Self-induction. (See Inductance.) Separation between overhead wires, 157 Series system. (See Thury system.) Service, continuity of, 3, 7, 42, 125 Single-phase transmission, 14, 369 Skin effect, 15, 71, 77, 84, 85 Sleet. (See Ice.) Snow and ice, effects of. (See Ice.) Spacing of overhead wires, 156 of lightning arresters, 192 Span, influence of, on cost, 39 length of, 3, 37, 38, 298, 304, 380, 390 Spans on steep grades, 256, 294 Spark-gap lightning arresters, 180 Sparking distances, 132, 160, 181 Specification for steel tower line, 386 for wood pole line, 375 Specific inductive capacity, 99, 130, 213 resistance of cable insulation, 214 Standing waves, 173 Steel cored cables, 73 masts or poles, 329, 330 towers. (See Towers.) wires and cables. (See Wire, iron.) Steep grades, lines on, 261 Stresses in wires due to wind and ice, 264 "String efficiency" of insulators, 137 Stringing wires. (See Wires.) Struts or props, wood, strength of, 323 UMMfV