The Effect of a Steel Core upon the Electrical Characteristics of Large Stranded Aluminum Conductors Citation Rich, Kenneth Whiting (1916) The Effect of a Steel Core upon the Electrical Characteristics of Large Stranded Aluminum Conductors. Bachelor's thesis, California Institute of Technology. doi:10.7907/rqqa-7621. https://resolver.caltech.edu/CaltechTHESIS:07152024-160116854 Abstract No abstract. Item Type: Thesis (Bachelor's thesis) Subject Keywords: (Electrical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Electrical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1916 Record Number: CaltechTHESIS:07152024-160116854 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:07152024-160116854 DOI: 10.7907/rqqa-7621 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16552 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 15 Jul 2024 17:06 Last Modified: 15 Jul 2024 17:07 Thesis Files PDF - Final Version See Usage Policy. 10MB Repository Staff Only: item control page

THE EFFECT OF A STEEL CORE UPON THE ELECTRICAL CHARACTERISTICS OF LARGE STRANDED ALUMINUM COlf.DUCTORS . .I:HESIS 1 CLASS OF 1916 THROOP COLLEGE OF TECI-IlTOLOGY THE S I S References Consulted: Bureau of Standards Bulletins, Reprints Nos. 80, 25a. Electrical Properties of Compound Wires F. F. Fowle, Electrical World, December 1910, January 1911. Constant Voltage Transmission Dwight. Electric Traction and Transmission Engine er ing. Sheldon & Hansman. EJ.e ctric Power Cona. uctors. Del I..Iar :Magnetic Induction in Iron and Other Metals J. A. Ewing . .Fender's .American Handbook for Engineers. Standard. Handbook for Electrical Engineers. nrFLUElWE OF STEEL CORE UPOH THE l~LECTRIC.AL CHARACTERISTICS OF LAJ.lGE ALUMIUU:NI STRAlJDEil CONDUCTORS. The determination of the electrical characteristics of a large aluminum stranded power cable with a steel core has been divided into the following sections in this paper: 1. A general description of the cable in 2. Tests for the permeability of the steel 3. An outline of the method followed in the question. core. tests, with the data sheets. 4. A consideration of the resistances of core and cable for direct and alternating E. M. Fs., including the skin effect. 5. A comparison of test results for inductances ·with those calculated b:,r formulae. 6. An appendix containin8 matter relevant to this subject. ., .L • The cable under test was a section of the stranded aluminum ca.ble used by the :Pacific Light & Power Corporation in transmitting power from their hydro-electric stations on Big Creek to Los Angeles two hundred and forty one miles distant. The system is three phase, fifty cycles, and is designed. to transmit 60,000 kilowatts at 150,000 volts. Fig. 1 is a photograph of an end of the cable showing the three layers of aluminum strands and the steel core in the center. Conductor Strands Cir. Mils outside Diam. Aluminum 54 605,000 . 9(16 inches :k" / •;;, 7 78,500 . 3125 inches Wei ght per mile Specific Gravity (assumed) Ratio of Conductor area to total area Ratio of wts. to total wt. 2940 lbs. 7.85 88.50 % 72.5 ~ I ._, 1118 lbs. 2.68 11.50 % 27.5 '704 Steel It is of interest to compare various qualities of this cable with those possessed by solid copper or solid almninum cables. The ratio of actual weight to that of a solid copper cable of the same size is 0.367, and to one of solid aluminum 1.22. If spec'ific conductances be taken for copper 2. 10. 37, f or steel 74. 0 , and for alu.~inum 16.72, t he actual conductance co mpared. with that of a solid co pp er cable is 56.48%; of a solid aluminum cable 91.11%. The ratio of a mile ohm of this cab le to that of a solid copper cable is 0.651, and to one of solid aluminum 1.34. The wire is wouna in three l ayers outside the core. In the outer layer t here are 24 strands with one complete t wist ea ch 18 inches, t he next layer has 18 strands and completes a t wist in the opposite direction every 10 inches, t he layer next the core is made up of 12 wires with a still greater twist, the direction being a.gain reversed.. PERMEABILITY OF THE STEEL CORE. To determine the permeability of the steel core the method outlined in Pender's American Handbook for Electrical Engineers, page 913, was follo wed, us ing a Leeds and Northrup Astatic Galvanometer. For the standard solenoid mentioned in this method a portion of a permeameter, the dimensions of which were accurately known, was used. The length of the sample in comparison to its cross section made end co r rections unnecessary. The set up is shown in Fi g. 2 and the results are plotted in Fig. 3. Calculations are all given below. B = .4 71 n, N1 A1 I, n A a AH A N, ==- turns in primary of solenoid per cm. length (axia l) n,= total turns in secondary of solenoid. n = fl !I Tl test s ample. A,= mean cross sectional area in cm. sq. of primary of standard if secondary be on outside, or mean cross section of secondary if secondary be on inside. A = Cross section in cm. s~. of test sample. a= mean cross section of ma gnetizing coil of test s ample. I,= current thru primary of test sample. D = deflection when current reversed. B = flu.xdensit y in test sample corresponding to deflection D when current is magnetized coil on sample is reversed. H = ms.gnetizlnq force in sample corresponding to :fluxdens i ty B , Av. diam of core .312 inches, n,= 200; A = 1.67 sq. cm.; 1 K= n = 36; .47Tn,N,A, _ n A - N,= 31.5; A= .0764; .4 x]Tx 200 x 31.5 x 1.57 = 4820 canst. 36 :x: .0764 a _ A H _ .109 - . 0764 H _ 0326 H _ C H .A . 0764 - • - = K I, B D 4820 X .49 4820 X .59 4820 X .79 4820 X 1.00 4820 X 1.29 4820 X 1.62 4820 X 2.10 4820 X 2.28 4820 X 2,59 4820 X 3.48 2360 2850 3810 4820 6220 7810 10100 11000 12500 16800 .21 . 28 .35 .46 .58 .70 .89 .99 1.11 1.52 B for values used this term is negligible These values were used in plotting flux density curve. TEST PIECE H - .4 7T :N I·' 1:1 =- turns per cm. Deftection Current .12 .16 . 21 .23 .9 1.05 1.28 1.4 1.63 1.91 2.3 2.72 2.97 3.51 10.1 . 30 .42 .71 1.19 1.62 2.7 6.2 5. N = 360 _ 5.07 vr- B H B )l =H 1350 1800 2300 2600 3350 4700 7900 13300 18500 30000 5.74 6.7 8.16 8.94 10.4 12.2 14.7 17.5 18.95 22.4 235 269 282 291 322 385 537 760 976 1340 t' ! ~ ~ tJ ~ .ia..,. ~ ,...~ ~ ~ ~ V) t~ ... .,.,, t ~ .... t ·••:i?FP-,!~ :~_)·.:_T?7_·.-.--_:, - 1 ~1~·1 ; :- ···:~:r~·?.~~:-rl:-~T?F YT~·.: r: :. .,_ -~ : T • 1 .. • 1 - .,.. . - i._ --•·. 1 ?;r: -T~ ~r~FFt=T ·-•.,· l ···;··- ·· r- t -- ----- - - - 1 ·- ' ' j ,- .J 1 ' I -- - --1- ~ . l - - - - -- --· I .. i I I". : • ··+ I ! -- --- i . ... •·· -- ,- - ' i • --- -t-- ; - --i- ' . ' -· ---- - -- . ., _ - .. ' --· ------ I _i ·I i l I - . / -i _/ / / / / z.zooo / / l.000 / . / 1801)() ~ ~ --: 'I;) 14 i· · -· ~ ~ JZJ){)() ~ / 0(}{)() /:t)()() Z.OIJC _ _j 2.1)() 31)0 4DO $1() t),8 /,'() DIJ() 70() Boo l ~ Pt:l"mia.b1.'Yt.'ty : 0,1. P.4 I. 2. 1,4 •I. :ti • • /.8 be/le<! ti.on~,Ga/yq,;t:Jme~;, Saile Z..fi The permeability thus found is that lengthwise of the wire, whereas in the ~ctual tests for inductance the flux is perpendicular to the length of the wire. No method for investigating the permeability Cil'osswise appeared feasible, the high cost per foot of the cable precluded stripping the aluminum from any considerable length of the core. 6. OUTLINE OF METHOD USED DJ RESISTANCE .A:ND nmuCTANCE TESTS. The following method was used in determining the alternating current resistance and inductance of both core and o.~ble. A''fencerr was built somewhat more that a foot high and in the sh~.pe of a rectangle 366.3 ft. long and of a variable width. The wire was stretched upon the boards forming the sides of the rectangle. Strong clamps of large cross section made good contact between cable and instrument connections. The photo- graphs, Fig. 4 and Fig. 5, give a good idea of the whole set up. All instruments used were carefully calibrated and curves were used to correct the readings obtained. Readings were taken of volts, amperes, watts, frequency and temperature. In some runs frequency and temperature remained. constant, in other c@.ses simultaneous readings were taken of all five values. The a. c. resistance was determined for each reading from the relation W=I£R . These values we.re:. then corrected for temperature based on the resistance at 20 ° C. Temperature corrections were based on a solid aluminum conductor, the core carrying a very small proportion of the total current as appears later. The resistance per thousand feet of cable was next found and the arithmetical average 7. of the separate runs taken as the true resistance per thousand .f eet. Finally, these values were reduced to resistances for the actual lengths tested for use in calculating the inductance. The impedance was found from the equation E = S I and the average impedance based on the ari thmetical average taken for the run. The value of the reactance came from the equation X=--YZ~ R'- and from these values knowing the fre- • quency the inductance was easily calculated from the relation X =21TfL, For watt readings the Kelvin Balance was used, a previous run with a small wattmeter and current transformer proving unsatisfactory. The Balance was checked with direct current and found accurate. The system of averaging resistances and impedances was used onl~l after complete calculations from the separate readings were worked out and this method shmvn to be accurate. The direct current resistm1ces were taken with laboratory standard instruments and a storage battery. The data sheets follow. 8. STEEL CORE, 15 Foot Spacing Length of Side 22.75', Total Length 86.5' . of Cable D. C. Measurements R. 112.6 78.8 59.8 47.9 40.5 113.1 112.9 48.0 40.5 .977 .682 .518 .413 . 348 .978 .9775 .4125 .348 .11525 .1155 .1154 .1158 .1164 .1155 .1155 .1163 .1164 Averaged. c. resistance .1158 ohms for the length tested, or 1.339 ohms per 1000 ft. A. C. }.ieaaurements Volts Corrected 4.61 5.27 5.76 6.12 4.19 4.69 5.00 5 .29 5.65 4.92 5.45 5.82 5.05 Amperes 4.64 37.2 5.40 42.9 5.98 6.40 4.12 4.72 Current Squared 1384 1840 2190 46.8 49.6 33.0 2460 Watts 23.4 165.0 228.7 268.75 308.75 138.75 171. 25 203.1 228 .7 260.0 197. 5 253. 7 292. 5 16.9 211.25 13.2 18.3 21.5 24.7 11.a 13.7 37.8 1089 1429 5.09 5.41 40.5 1640 16.25 43.2 5.85 5.00 5.62 6.06 5.15 46.1 1866 2125 1648 2061 2362 1722 18.3 20.8 15.8 20. 3 40.6 45.4 48.6 41.5 9. Watts Corrected Temperature Cycles He s istance 92.5 92.5 92.0 92.0 91.0 91.0 90.0 90. 0 90.0 82.0 81 .5 81.0 80.0 50.25 .1192 .1243 .1227 .1255 .1274 .1198 .1238 .1226 .1233 .1198 .1231 .1 238 .. 1227 11 Tl lT II TT II Tl ff 50 .3 II 50.4 50.6 Temp. Re sist . Corrections Corr. .0068 .0071 .0069 . 0071 .00 69 .0065 .0064 . 0066, . 0064 . 0038 .0038 . 0037 .0035 Impeda nce .1124 .1172 .115 8 .1184 .1205 .1133 .1174 .,1'160 ,: ,. 1169 .1161 .1193 .1201 .1192 .1247 .1259 .1278 .1290 .1249 .1249 .1257 .1252 .125g .1232 .1238 .1247 .1241 Average impedance .1254 ohms. 0 Average a . c. resistance .1175 ohms, at 20 Cent. 11 11 TT n per 1000 ft. 1. 360 ohms . Z 'Z. - Rz.:: .00192 X = . 0438 L= 138.6 millihenrys RESIS~AlTCE 0] 1 CABLE TO DI HE CT CURRENT Milli Volts 1 3.'7 22.0 26.4 40 . 5 27.7 20.8 17. 0 14. 25 Mil li .Amperes 637 102.0 1 230 1890 1285 962 ?86 660 Re sistance . 0215 0 .02158 .0 2145 .02143 .0 2158 .0 2160 .0 2163 .02160 Temp era ture 28.5° C. Average resistance a t 20° C. of 750.5 ft. of cable is .02077 ohms . Resistance per 1000 ft. is .0 2767 ohm~ . Resist&n ce per mile,0.1461 ohms at 20 C. Weight is 592.9 pounds per mile -ohm. If aluminum only,d. c. resista,nce would be .0281 ohms pe r 1000 ft. 10. CABLE 9 ft. SPACilJG Volts Volts Corrected Ampere~_ Amperes Corrected Current Sauared Watts 9.40 5.15 • 6.05 6.65 7.80 8.40 8.70 9.50 9.00 8.70 8.15 6.70 10.39 5 .67 6.65 7.31 8.57 9.21 9.53 10.40 9.85 9.53 8.94 7.37 130.0 77.0 89.5 96. 0 109.0 118 . 7 1 21.3 131.5 123.5 120.0 113.0 85.5 127.6 75.3 88.0 94.25 107.7 116.5 118 .6 129.1 120.9 117.6 111.3 84.0 16280 5660 7740 8880 11600 13580 14060 16700 14600 13820 12400 7050 31.7 11. 2 15.5 17.6 22.7 26.0 27.8 32.3 28.9 27.3 24.3 13.7 Watts Corrected. Resis- Temper- Temp. tance ature Corr. Rest. Corr. Impeda.nce 396.0 140.0 194.0 220.0 284.0 325.0 347.7 406.0 361.5 341.0 304.0 171.2 .0243 .0247 .0251 .0248 .0245 .0240 .0237 .0244 .0241 .0240 .0245 .0243 92.0 91:;0 86.0 85.0 83.0 82.5 82.0 80.5 80.0 79.0 78. 5 78. 0 .00133 .00136 . 00106 .00105 .00083 .00081 .00080 . 00072 .00068 .00062 .00060 .00058 .0230 .0233 .0237 .0237 .0237 .0232 .0229 .0237 .0234 .0234 . 0239 .0237 .0815 .0754 .0756 .0776 .0796 .0791 .0784 .0806 .0798 .0794 .0790 .0879 Average a. c. resistance at 20° C .0235 Corrected for length tested .0237 ohms. Resistance per 1000 ft. a t 20° C. . 03131 Frequency 50.25 cycles constant. Average i mpedance= .0787 z z. - X2 .005632 X,,.--=- .0750 L -= 237. 6 millihenrys induct'ance 11. CABLE 11. 8 ft. SPACING Vol ts Corr. Amperes Ampe re s Corr. Curren t Squared Wa.tts Volts 8 .12 8.45 8.78 9 .20 9.50 9.55 9.90 8 .82 9.26 9.69 10.375 10.80 10. 88 11.40 10 3 . 5 109.0 116.0 124.0 130.0 132. 0 136. 5 102. 107.8 -114.0 121.5 127.6 129.6 133.8 10400 11620 13000 14760 16280 16810 17900 20.0 22.3 24.7 28.2 31.1 32.6 34.0 Watts Corr. Resistance Temper-_ Temp. Corr . ature Rest. Corr. I mp eda nce 250.0 278.7 308. 7 352.5 388.7 407 .5 425.0 .02430 .02398 .02375 .02388 .. 02388 . 0242 7 .02374 78.5 . 02368 .02337 .02315 .02327 .02327 .02365 .02314 .08 64 7 .08590 .08500 .08539 .08464 .08394 .08521 Tl !I If n Tl 79.0 . 00062 . 000 61 .00060 . 00061 .0 0061 .00 062 000060 Average a. c. resistance: .0234 ohms at 20°0 c. Average - a. -c. resistance pe-r 1000 ft~ e.t·.· ;w c.= .03093 Correcte_d va:Lue for length t estea. = . 02389. Frequency 51 cycles constant. Average impedance = .08522 ohms z2. n.:z. =- . oo 6691 X-= .08179 L: 255.3 millihenrys inductance 12. CABLE 15 foot S:PACIITG Volts Volts Corr. 7.95 8.60 8.85 9. 35 10.25 11.97 12.30 11.05 7.80 8.20 8.58 9.50 9.835 10.57 10.52 12.15 1 2 .52 11.14 8.40 8.91 --- Curr. Squared Watts Amperes Amps. Corr. 94.0 103.0 107.0 115.0 114.0 133.0 137.0 121.0 91.0 98.0 92.3 101.5 105.7 113.l 112.2 130. 05 134.1 118.3 89.3 96.4 8520 10300 11180 12790 12590 16915 17970 14000 7974 9292 16.5 20.05 21.3 25.0 24.6 33.3 34.8 27.9 15.9 17.9 Watts Corr. Resistance 206. 23 250. 60 266.16 312.50 307.50 416. 20 435.00 348.74 198.74 223.73 .02421 .02433 .02381 .02443 .02442 . 02460 .02421 .02491 .02492 .02408 Temp. Corr. Rest. Corr. Impeds.nce .00020 .00021 .00020 .00021 .00021 .00021 . 00020 .00021 .00020 .00020 .02421 .02412 .02361 .02422 .02421 .02439 .02401 .02470 .02471 .02388 .09286 .09359 .09304 .09346 .09366 .0934 3 .093~7 .09416 .09407 .09244 -- Avera ge a. c. resistance at 20° c. = .02421 ohms Average a. c. resista nce per 1000 ft.= .03173 ohms Corrected value for length tested= .02409 ohms Average i mped ance: .09341 Frequency 50.9 cycles, Temperature 71.5 0 constants. z" - RlL =- . 000145 .09022 L ~ 282.1 millihenrys inductance X : 13. CABLE, 18 foot SJ?ACIHG Volts Volts Corr. .Amperes 7.67 8.20 9.15 9.28 9.80 11.25 11.85 12.60 11.25 13.10 8.23 8.91 10.27 10.46 ll.ij6 11.33 11.925 12.96 11.33 13.45 85.0 92.5 106.0 108.0 116.0 119.0 126.0 135.0 119.5 141.0 Watts Corr. Resistance 167.5 208. '7 273.8 285.0 332.5 345.0 387.5 441.3 345.0 480.0 .02408 .02526 .02503 . 02506 . 02557 .02533 .02538 .02522 .02559 .02522 Amps. Corr. 83.4 90.9 104.6 106.7 114.0 116.7 123.6 132.3 116.l 138. Current Sq_ua:red 6956 8268 ;] 10940 11370, 13000 13620 152'70 ;t '7500 ·,13480 19040 Watts 13.4 16.'7 ·-, . 21.9 ''22. 8 26.6 27.6 31.0 35.3 27.6 38.4 Impedance Corr. Rest. Corr. .00085 .00086 .00085 .00085 .0008'7 .00086 .00086 .00086 .00086 .00086 .02323 .02440 .02418 .02421 .024'70 ,0244'7 .02452 .02456 .02473 .02436 .09868 .09802 .09690 .09804 .09868 .09'708 .09648 .0979'7 .09844 .09746 Temp. Average a. c. resistance at 20° C. = .02439 opms. 11 11 TT II n n TT per 1000 ft.= .03171 Corrected value for length fasted = .02428 ohms. Frequency 50. 8 cycles, Temp. 82° Constants. Average impedance~ .09'7'7'7 ohms. zz_ R:z..= .008969 X -:: . 094'7 L ~ 296.7 millihenrys inductance. 14. CABLE, 20 foot SPACING Volts Volts Corr. Amperes Amps. Corr. Current S~uared. Watts 8.25 9.70 12.30 13.45 12.00 12.85 8.00 8151 8.94 9.10 8.99 11.10 12.51 13.85 12.177 13.15 8.63 9.35 9.94 10.20 91.5 111.5 129.0 142.5 126.0 136.5 88.3 95.2 101.4 104.0 89.9 110.0 126.7 139.5 123.5 133.8 86.8 93.5 99.9 102.4 8082 12100 16050 19460 15250 17900 7534 8743 9980 10480 1~.8 25.2 33.9 40.9 31.8 37.3 16.15 18.90 21.50 28. 25 A Watts Corr. Resistance Temp. Corr. Resist. Corr. Impedance 210.0 315.0 423.8 511.2 397.5 466.5 201. 9 236.2 268.8 282.5 . 02598 .02603 .02640 .02627 .02606 .02606 .02680 .02710 .02693 .02696 .00143 .00143 .. 00145 .00144 .00143 .00143 .00147 .00149 . 00148 .00148 .02455 .02460 .02495 .02483 .02463 .02463 .02533 . 02561 .02545 .02548 .10000 .10090 .09874 .09930 .09860 .09840 .09942 .10000 .09950 .09962 Average a. c. resistance -= .02499 ohms. 11 n n " per 1000 ft.= . 0322 ohms. Corrected value for length tested.-=. 02441 ohms. Frequency 50.8 cycles, Temp. 91 constants. Average impedance= .09945 ohms. ° Z.z..- R' =- .009294 X =- .09638 L = 302 millihenrys inductance 15. RESISTANCES The resistances to alternating currents for both core and cable are plotted in Fig. 6 and Fig. 7 and the averages indicated. The increase in cable resistance to alternating currents over that to direct currents is 14.9 %. Ken..11elly in a paper entitled. ''Experimental Researches on Skin Effect in Conductorsn :published. in the Proceedings of the American Institute, August 1915, makes the following remarks. Current distortion is due to three classes of effects: 1. Skin effect proper. 2. "Spirality" effect in stranded conductors. 3. nproximity 11 effect due to the spacing of the return conductor. The "spiralityn effect adds somewhat to the skin effect in the case of stranded conductors. 11 The proximi ty 11 effect is imperceptible for frequencies under 100 a.nd spacings above 20 ems. Pender's American Handbook makes the following statement: ''Change in resistance due to skin effect is always relatively much higher than the change in inductance. The following theoretical formula for skin 16. ! , effect was taken fro m the Standard Handbook. r = R ( 1+..l.( 2 7r f 1 )l ) _ ( 12 R K 10 q ) where, 1 180 r - a. c. resistance R - d . c. resistance 1 = length in ems . f = frequency in cycles per. sec. p = permeability usinc; this equation to s olve f or the permealJili ty of the core, p = 58 fo:r a current value of around 43 amperes flov'lin g in the core. So lving the same e quati on for the cable gives a value of 3 . 8 as an equivalent permeability for the whole cable. Penders American Handbook gives the following e quations for calculating skin effect. 7f fl. f where y, f and R X ;; .02768 , - - - - R have the same significance as above and xis a number which determines values for skin effect conditions. K 1 and K2 , constants for K1 and K2 are listed in a table. So lving for skin effect in core, the alternating current resistance to be exp ected would. be l.0144X 1.339 or 1.360 ohms per 1000 ft., a value whi ch is identical with the test result. Using the same equation for the cable the so lution e;ives . 0312 ohms a . c. resistance per 1000 ft. 17. of cable, thus checking closely the value .0316 the test {:tverage. Pender's American Handbook also gives the following equation for calculating the added effect of inductance caused by skin effect. -sf= L + .0152 f )l k2.- ll, L 1 and L being total inductance and inductance minus skin effect respectively, and )l and Kz. having the same significance as stated above. The second term on the right then is the added inductance due to skin effect. This term is so small as to be nearly negligible in these tests having a value of about .04 millihenrys per 1000 feet. The percentages of current carried by core and aluminum may be approximated by considering them as two circuits in parallel and solving by means of comparing admittances. Since the resistances and reactances are known the conductances and susceptances are easily f ound, and hence the admittances. This method gives for the admittance of the core 0.688 ohms and for the aluminum 7.84 ohms. !filhe core should then carry 8.78% of the current and the a luminum 91.22% The ratio of core a. c. resist ance to cable a. c. resistance is 43.1 : 1. If copper took the place of aluminum, based on 18. 61% conductivity of aluminrun the cable resistance would be .. 0193 ohms per 1000 feet and the ratio of core resistance to cable resis.tan.ce in that case would be '70.54 : 1. IlrDUCTAlWE. The test results of inductances are shown in Fig. 8. For comparative purposes the inductance was a.lso calculated by formulae. The first formula used was that of E. B. Rosa, of the Bureau of Standards, and is for solid conductors. The formula is given below. L = 4 ~ (a+ b ) log 2 ab "I! - b log (b+d) - 7 4 .:, a log (a+d) (a+b) + 2(d-+p) ~. L = inductance in centimeters. a = length in centimeters. b:width or spacing in centimeters f = radius of wire in centimeters :: 1. 205. d =-1 a,,_+ bi. Spacin.cr a b d L 9I 11,175 274.5 11,200 263,000 12' 11,175 366.0 11,200 273,000 15 I 11,175 456.5 11,200 284,500 18' 11,175 548.0 ilil,200 296,000 20 1 11,175 610.0 11,200 302,000 19. TI. B. Dwight on p~ge 115 of the appendix to his book entitled "Constan.t Volt~ge Transmission 11 gives the follovdng formula calculated for a 61 strand one metal conductor. L = 7 41.13 log 10 __ 2_.5_9_0_x...;.S;..____ x d S = spacing in inches d = actual diameter of cable L = inductance in henrys per mile Spacing in feet Inches Inductance in millihenr~rs 9 108 262.5 12 144 277.8 15 182 290.9 18 216 301.25 20 240 307.7 The inductance is that for the numlJer of feet of cable in the rectangle, the formula being develo ped for a line of infinite length. It will be seen in Fig. 8 that the test result coincides exactly w~th the inductance calculs ted by the Bureau of Standards formula at the 20 foot spacing only. Fi g . 9 shows the percent differences between formulae and test results at different spacings . 20. As a result of these tests it is recommended that t : : ie value of the a. c. resistHnce used in calculations of the Big Creek Cable be that of the d. c. resistance increased 14.9 %, and that the reactance values be those found by the standard methods. 21. APPE1IDIX When preliminary work was under way, ana. before any results were obtained it was believed the inductance would plainly show an upward slope as the current increased due to the flux in the iron core. At this time the discussion led to the development of a formula whereby the permeability could be calculated., lmowing the inductance, currents in core and aluminum and the physical dimensions. DulEond Credit is chiefly due to Mr. J. W. of the Class of 1916 for developing this formula which is hereby given as possibly being of value future. 22.. in the Let ¢., :: flux due to current in iron 11 11 11 02. ::. " " aluminum r1 d - ~ 'Pi:. l i 1 p+ 2i1 x log 2 r, dis the spacing of cable i, is the current in the core ii. is the current in the aluminum r, is the radius of the core r 2 iS the radius of the cable p is the permeability 1 is the length ¢2.is the same as flux if the tube were solid, minus the flux due to aluminum core. r~ 1 1 ~z. - "2' i.z r2.-:z. r,. - 2 i~ I r '2.. I r.,_r~ 2. rt= d - r i.. r.z.. d - r, ~, but solving for rj.L = Let K,= 2 log ( K 1 .:: 2 log 1 . r1 2 d 1-i-i,a -r r 1 '); 1 y T ,_ ( 1 n+- 1 K I n Kr Ltotal =- 2 1 (n + l ?- 1 - K ' = ( 1 + 2 1 og r1 r.i:+7:" ',.z. ... d - r. r ~ K = (- + 2 log ~ 2 K2. i Z.:+ KI i I+ i, b . 2 - i a I curren t I z 1. 1+ i :z. th en 1.. 1 =, n+l n ,)1 1 r1 = 'P rl _ 1U - - r, 2' i, )1 = 'Pl = 2 1 i 1b 1- r, I 2 r'Z.. r, 1 i ---+ )l = a ¢i.+-b; d 1 • 2 i, log t "2" i, p. +- - r2- I %, 1 il-+ 2i2- log 2 d + 2i-z. log , + Z:4 2 r2. Let i z.-- n i,_ and the total n I ana.~ i -z.::. n+ 1 I b 8n+2. - I a l K+ J• b I ls total c·urrent. z n+1 I 2 n+ 2 - I a ) 1 + 2 d ( n K + l K' b )• {n + 1 z n+ 1 ,+ 2 n+2 - Ia ) 2. 3..