None
V potential at anj' i>oint on a line (volts Z). Vi potential at a point 1 km. beyond the reference point (volts Z). Va potential at generator end A of a line (volts Z). Vb potential at motor end H of a line (volts Z). Vc f)otontial at a point (■ of a line, wh(Te the electrical con- ditions are known (volts Z). Vp potential at a point P on a line (volts Z ). Vx potential at an unknown point A' on a line fed from both ends (volts Z). 340 ARTIFICIAL ELECTRIC LINES ViT, Vb potential at junction N and leak N of an artificial line (volts Z). Vejfi VeB potential at position of junction N and of leak N, resper- tively, on a conjugate smooth line (volts Z). V potential at a leak in the presence of its admittance (volts Z). V'lf F'j, V'l . . . Fourier sine component amplitudes of a complex ha^ monic voltage wave (volts). V"i, V'"j, F"i . . . Fourier cosine component amplitudes of a complfi harmonic voltage wave (volts). VVi, Vr2j Vrs . . . Fourier resultant component amplitudes of a complex harmonic voltage wave (volts). Vof Vi, Fj, V| . . . Fourier resultant r.m.s. component amplitudes of a complex harmonic voltage wave (volts). ^ phase angle of a condenser (defect from 90**) (degrees). W maximum cyclic energy in a.c. circuit or conductor (joules). Wa active or real component of W (joules). Wg reactive or imaginary component of W (joules). Wm = 2Tf, maximum cyclic magnetic energy in a.c. circuit or coft- ductor (joules). X reactance of a coil or of a line (ohms) ; also in the theorr of skin effect, the radius of a conducting wire (cm.). X Cartesian coordinate on X axis; also distance along & line from a reference point in a down-energy direction (km.); also radius of a point in the cross-section of a wire (cm.). Y dielectric admittance of a line (mhos Z). Y/Ai YgA line admittance at A, with motor end freed and grounded (mhos Z). Y/Ai YgA line admittance at A, with motor end freed and grounded (mhos Z). Y/Ai /h linear dielectric admittance of a line after regular leak loadini (mhos Z). Y/Ai /h linear dielectric admittance of a line after regular leak loadini (mhos Z). Y/Ai /h linear dielectric admittance of a line after regular leak loadini (mhos Z). Y/Ai /h linear dielectric admittance of a line after regular leak loadini (mhos Z). Y/Ai /h linear impedance of a coil or of a line conductor (ohms Z). Y/Ai /h linear impedance of a round wire due to skin effect (ohms/wire km. Z). Y/Ai /h linear impedance of a round wire due to skin effect (ohms/wire km. Z). Y/Ai /h linear impedance of a two-wire line (loop ohms Z). Y/Ai /h linear impedance of a two-wire line (loop ohms Z). Y/Ai /h linear impedance of a two-wire line (loop ohms Z). Y/Ai /h linear impedance of a two-wire line (loop ohms Z). Y/Ai /h linear impedance of a two-wire line (loop ohms Z). Y/Ai /h linear impedance of a two-wire line (loop ohms Z). Y/Ai /h linear impedance of a two-wire line (loop ohms Z). Y/Ai /h linear impedance of a two-wire line (loop ohms Z). Y/Ai /h linear impedance of a two-wire line (loop ohms Z). Y/Ai /h linear impedance of a two-wire line (loop ohms Z). Y/Ai /h linear impedance of a two-wire line (loop ohms Z). Y/Ai /h linear impedance of a two-wire line (loop ohms Z). Y/Ai /h linear impedance of a two-wire line (loop ohms Z). Y/Ai /h linear impedance of a two-wire line (loop ohms Z). Y/Ai /h linear impedance of a two- Line (ohms) loop km. Zr impedance of a motor-end load (ohms Z). W = 2irf angular velocity or angular frequency (radians per sec.). LIST OF SYMBOLS EMPLOYED 341 M« = 4/ angular velocity or angular frequency (quadrants per sec.). 2oo — \^l/c limiting value of surge impedance ignoring losses (ohms). ztj 2s in theory of composite lines, surge impedance of successive sections (ohms Z). zo, surge impedance of a line section after loading (ohms), oc sign of infinity. Z sign of a complex quantity or of the slope of a complex quantity. \a size of the complex quantity a (numeric). a slope of the complex quantity a (degrees or radians). ^ sign for "cycles per second." sign for ohms impedance, r) sign for mhos admittance. ^ sign for "nearly equals," or approximate equality. / sign for farads capacitance. tif sign for microfarads capacitance. myif sign for millimicrofarads. MMf sign for inicromicrofarads. logh hyperbolic logarithm to base c. log common logarithm to base 10. hyp. contraction for "hyperbolic radian." r.m.s. contraction for "root mean square." B.C. contraction for "alternating-current." c.c. contraction for "continuous-current." 4 INDEX Abhenry or C.G.S. magnetic unit of Becker, G. F., 120 inductance, 143 Active conductance, 129 resistance, 125 Actual velocity of propagation, 153 Addition and subtraction of vectors, 117 Admittance, linear wire, 149 Aggregate conductor resistance R of a line, 16 dielectric leakance Goia, line, 1 6 Ahlbom, G. H., 213 All-capacitance artificial line, 312 All-inductance artificial line, 312 All-resistance artificial line, 312 Alternating continued fractions, 327 Alternating-current artificial lines, fundamental properties of, 164 simple circuits, 124 Anderson bridge, 234 Angle hyperbolic of a line, 16 Angles, real circular, 6 real hyperbolic, 6 Apparent velocity of propagation, 152 Architrave admittance, 242 Artificial line, as frequency filter, 316 definition of, 1 homologous types of, 310 principal purposes of, 1 sections, unrealizable, 167 various types of, 309 lines and duplex telegraphy, 2 historical outline of, 2 Attenuation nmstant of a.c. line, 151 of a line, 16 factor, normal a.c, 156 normal of a line, IS Axis of imaginaries, 114 of reals, 114 Bedell and Crehore, 127 Bessel functions, 147 Bifurcating composite lines, 275 Bouton, C. L., 123 Cable circuit, pure, 156 Campbell, A, 236 Campbell bridge, 234 Campbell, G. A., 110, 302 Campbell mutual-inductance bridge, 235 Cantilevers, 252 Casual loads, 254 Changes in architrave formula with casual loads, 276 Characteristic resistance of a line, 19 Chrystal, G., 123 Circuit, distortionless, 154 Circular angles, real, 6 and hyperbolic formulas list of, 3i23 Clerk-Maxwell, 141 Cohen, B. S., and Shepherd, G. M. B, 181 Cohen, B. S., 233 Complex liarmonic waves, 192 hyperbolic angles, 119 quantities defined, 113 Composite line, a.c. Example, 261 of impedance terminal loads 268 architrave and casual loads, 276 impedance, second method, 250 attenuation in miles, 286 definition of, 15 equivalent II of, 249 intermediate leak loads in, 272 impedance loads in, 272 345 Composite line, leak loads in, 271 transformer in, 278 lines, 240 bifurcating, 275 line-current distribution over, 247 position-angle distribution over, 247 Condensive susceptiveness, 129 Conjugate smooth line, 62, 64 Constant, attenuation, of a line, 16 continued fractions, 327 propagation, of a line, 16 wave-length, 152 Constants, linear of a real line, 15 Contacting polygons, internal and external, to equiangular spiral, 169 Continued fractions, alternating and constant, 327 ("correcting factor for a leak load, 111 factors for T and n sections, 91 Cosine of complex hyperbolic angle, 120 Cosines of real circular and hyperbolic angles, 9 ( -rab addition or perpendicular summation, 194 Cunningham, J. H., artificial line design, 5 Current at point along line in terms of position angle, 39 Cycloid, 136 Depression factor of a leak load, defined, 111 Design and construction of artificial a.c. lines, 196 Detection of even-frequency harmonies, 319 Detection of even-frequency harmonies, 319 Detectors of frequency variations, artificial hues, 322 Distortionless circuit, 154 Distortionless circuit, 154 Distortionless circuit, 154 Distortionless circuit, 144 Double surge impedance, T and U sections of, 211 Division of vectors, 118 Double surge impedance, T and U sections of, 211 Douglas, J. F. H., 292 Drysdale, C. V., 158, 224 Drysdale-Tinsley potentiometer, 222 Duplex telegraphy and artificial lines, 2 Even-frequency harmonics, detection of, 319 Equiangular spiral, 157 polygon, 169 Equivalent circuits defined, 89 of composite line, 249 T defined, 89 Estwick, C. F., 215 Exponential case, 45 Exponentials, geometrical interpretation of, 12, 13 External contacting polygon to equiangular spiral, 169 linear loop and external linear wire inductance, 141, 142 Factor, normal attenuation, of a line, 18 Feldmann and Herzog, 92 Fictitious impedance diagrams, 140 Filter of frequencies, artificial line for, 316 Fleming, J. A., 115, 116, 224, 235 Four-wire artificial lines, 56 Fractions, alternating continued, 327 Frequency filter, 316 Limitations of artificial lines, 239 measurements, 238 Frequencies, range of, 125 Functions, Bessel, 147 Fundamental and secondary constants of a line, 22 differential equation of line, complete solution of, 28 of a smooth line, 24 CJftti, B., 233 Croneral Railway Signal Co., artificial track circuit, 215 Generator end of line defined, 30 Graphical relation between potential and current spirals, 159 Gray, A., 110 Greenhill, A. G., 135, 157 Grover, F. W., 237 Half-wave and quarter-wave lines, 288 Harmonic suppression, under, 318 upper, 317 Harmonics, detection of even-frequencies, 319 sine and cosine, 193 Heaviside, O., 24, 29, 154, 163, 243 Herzog and Feldmann, 92 Historical outline of artificial lines, 2 Homologous types of artificial line, 310 Huxley, R. D., 204 Hyperbolic and merger of composite line, 243 angle, linear, of a line, 16 angles, complex, 119 real, 6 Hyperbolic-cosine, infinite series of integers, 314 radians, 7, 16 -sine infinite series of integers, 314 Sections of artificial line defined, 166 Imaginary axis, 114 component of rectangular vector, 114 Impedance graphs On artificial and a.c. conjugate smooth lines, 184 model of a.c. line, 231 tests, 219 Inductive reactance, 125 susceptance, 129 Infinite series of numbers, hyperbolic-sine and cosine, 314 Infra. su rèe impedance, 51 Initial current at sending end, 161 Intermediate impedance load in composite line, 269 Intermediate, leak loads in composite line, 272 Internal contacting polygon to equilateral spiral, 169 linear impedance of a wire, 148 loop and internal linear wire inductance, 141, 142 Involution and evolution of vectors, 119 Iterative resistance of a line, 19 Jahnke and Emde, 147 Laboratory tests of composite line, 280 Laws, F. A., and Pierce, P. H., 147 Leak loads, 254 regular, 304 Leblanc, M., 163 Lewis, L. v., 215 Line admittance, 48 current tests by ammeter, 232 impedance, 48 reactors, design of, 197 resistance in unsteady state, 51 Lines regularly loaded, 300 Linear and total hyperbolic angles of a line, 21 complex hyperbolic angle of line or section, 149 constants of a line, 15 dielectric leakance and admit- tance of a real line, 15 hyperbolic angle of a line, 16, 17 impedance, internal, of a wire, 148 inductance, 141 reactance, 141 resistance and impedance of a real line, 15 wire admittance, 149 capacitance, 144 impedance, 147 inductance, 142 reactance, 145 susceptance, 147 resistance and impedance of a real line, 15 wire admittance, 149 capacitance, 144 impedance, 147 inductance, 142 reactance, 145 susceptance, 147 surge impedance, 161 susceptance, linear, 147 Lossless line, 154 Lumpy artificial lines classified, 55 lines, definition of, 2 Magnusson, C. E., Goodorham, J., and Rador, R., 204 and Burbank, S. R., 204, 291 McCready, Harold, 213 Measurement of individual inductances, 233 Merger n, Motor end of line defined, 30 Muirhead, Alex, artificial submarine cable, 3 Multiplication of vectors, 117 Natural resistance of a line, 19 Negligible line leakage and reduction to Ohm's law, 36 Nominal II, defined, 91 defined, 90 Normal a.c. attenuation, 156 defined, 45 factor, 45 of a line, 18 graphs on a.c. lines, 167 spiral, 157 waves, 161 Number H, hypcotic-sino and rosino, infinite series of, 314 O Serions of artificial line defined, iri6 Orthogonal projection, 13 Osborne, 11. S., and IVmlcr, H., 144 Overhead aerial lines and their leaks, 28 Pender and Huxley design of reactor, 144, 205 and Osborne, 144 Perpendicular summation or crab addition, 194 line admittance on each side of a junction, 74 currents at junctions, 77 at mid-sections, 84 defined, 57 distributions worked out, 78, 79, 80 impedances at junctions, 69 at mid-sections, 84 junction potentials, 76 lumpiness correction factor, 73 potential at mid-sections, 83 powers at mid-sections, 84 section angle, 72 Plane vectors defined, 113 polar, 115 rectangular, 114 Plug contact piece, 209 Plural frequencies on a.c. lines, 192 Position angle, definition of Distribution on composite line, 245 angles, solution in terms of, 37 Potential in terms of position angles, 38 Potentiometer and voltmeter methods compared, 230 tests of c.c. 