MAGNETIC LOSSES IN TRANSMISSION LINE TIW‘IEPé HAVING A {.7133333E C MAGNETIC PATH SURROUNDING ONE CONDUCTOR ‘I‘II‘ecsis for the Degree. of M. S: .". .i;:.:as I‘v‘iwshed babbagh a .3 , 5 1‘” .' {V‘ .0_ can . .' . _ 5’. . ' I ‘ . R 39795" = . ' \ _ ‘J ' 4" 3w“. - 3'] .111”: ‘cd- '3'!" ' t w , . 7:; . '.P .I ' ‘ '7 K , ‘ .ij‘fizjfjg ”>.’(Q"T "'r ‘10?" 9 TWO" * " . :5, ' "" ' ' 'o _ Aw \. . s ' ~ ‘ ;~' I N‘.‘ ‘- ' o 11' ’. - 3."? I (V'I "..{‘i‘ , . 13‘ ‘ ('0‘...- -‘ 4“ 255* , ‘1(., _‘,~ ' .t' 1‘. ,. ' ‘0 ‘.- (ft-r. d # II; :'{.:. § M j} 451:; o .‘ ’. Wm, “ k 7"; d’. L. :4 4m .: fi?‘ ‘9' ‘1 ' ?:'I'r". 5.1. p u .chy, .. k,L ‘ 21“". .,_fi . ‘2 . | I 0 ' r ‘ + " r- -. “ ' ‘ . _ .¢ ‘;. ‘I ~ 'v'.‘ . . .. u... r39?“ ;; J‘ .‘M" ‘1 4‘! "IKQ ‘I ".H- ‘ 1 - . . fn" 4! > :‘\?:;76. "fi: .- u;V I" “'Ivc" ‘ M ,»I ‘. f, _ I. . ‘ . . ' "- ' . U" L A.‘ 5;" t 1‘?!- :- " ~ ’ a. .- WM . . - 2 - ,7 ,. . ’vI - I .d ‘ ' I , - I I k ’u _ . fl- "2 ' I ' I . W . IL“ I ‘4‘ ‘ V ‘ 'I ‘ A. 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' .. ~ , ”*",fi":2 ‘1’}; i: ‘4‘ ‘3 '11OI’ (:4 II " ';7-‘I~,." ‘x .' ’ k I..- III) 5%: I III 7“" £-$¢§§’«3 Whiz": .‘ J I x ‘I " I ‘r‘ h“ IJVI‘I f; '13::229422‘2-aa-eiw s; L. . .. “be.” #2,. rfi’ :g' I' MAGNETIC LOSSES IN TRANSMISSION LINE TOWERS HAVING A CLOSED MGNETIC PATH SURROUNDING ONE CODTHJCTOR. A Thesis submitted to The Faculty of the MICHIGAN STAB COLLEGE BY Elias Hershed Sabbagh Candidate for the Degree of Master of Science. June 1928. THESIS 10343—1 TABLE OF COIII‘EIJTS Introduction Acknowledgent The Problem Apparatus Used The Geometry of the L00p Theoretical Solut ion Reasons Why Losses in Legs a and b are Unequal Procedure to Obtain Results from the Osci negraphic Readings Sample Computati on Data and Results Graphs of the Distribution of Losses on the Legs of tie L00p Oscillogram of the Induced Currm ts on the Three Legs Result 3 fr an Expe rimmt Deduction of Laws General Formula ind Check 10 27 29 52 37 74 75 81 84 The loop, line, and exploring coil. INTRODUCTION Many transmission lines have one conductor passing through a closed 100p of structural steel. Such is generally the case of lines in hilly countries, or Where three or more transmission line circuits are carried on the same towerIstructure due to high cost of right of way, of transposition towers for double circuits, etc. Up to the present day it is customary in the design of a line, not to pay any attention to the magnetic losses in the towers, but for the sake of determining the facts we carried out such an investigation. Therefore, it is the purpose of this paper'to investigate the losses in the 100p and derive a formula by which itwnould be possible to compute the losses due to any current at any distance. Although those losses do not amount to a great deal in one single tower, they should be taken into consideration when the line is long and more particularly when it is under much load. The amount of magnetic losses then is not negli- gible. The losses causing heating in the structure of substation are similar to the losses in theczlosed loop of the tower. .at some points the heating iszniximum as illustrated at various points of the legs of the loop. ACKN G’iLEDGlILIT The.author desires to express his indebtedness to the Commonwealth Power Corporation of lfichigan for furnishing the necessary tower 11—. rts, to the Board of Water and Electric Light of the City of Lansing for lending transformers and supplying cables capable of carrying high currents. The.author wishes also to express his appreciation to the staff menbers of the Electrical Engineering Departuent of the Midnigan State College for the suggestions and help received from them, and to Mr. F. Mitchell for his collaboration. MGNETIC BCBSES IN MSMISSIW TOWERS HAVING A CLCBED MAGNETIC PATH SURRCUNDING (NE CONWCTCB The Problem: Hany transmission lines have one conductor passing through a tri- angular, rectangular or polygonal closed magnetic path formed by the structural steel of the tower. It is the purpose of this paper to determine the amount of the iron losses in the tower caused by induction. The triangular shaped ungnetic path was chosen as it is the most comnonly used. Procedure: A triangular shaped loop of structural steel was erected. A three phase transmission circuit was built. One conductor of the line went through the center of the loop, while the two other condictors were on the right and left side. The three conductors were in the same horizontal plane and at the same distance from the steel. many exploring coils of known nunber of turns were wound around the three legs of the loop at intervals. The induced voltage in each soil was measured and recorded. By this means the fluxes induced in the loop at different points due to different currents were calculated. By using Steinmetz' and the eddy current formulae the different losses were cal- culated and graphs plotted. Apparatus used: Due to the limitation in the available apparatus many schemes were planned for obtaining enough current in the line and for measuring the losses. It was first thought that by means of current transformers it would be possible to obtain any desired current in the line. The secon- dary coils of three transformers were used as primaries, and the primary sides were connected to the line. Although a large amount of current was flowing in the secondaries. (used as primaries) of the current transformers not enough current appeared in the line. This scheme was rejected after trying each of the following connections, Delta-Delta, Wye-Wye, Delta-Wye, and Wye-Delta. Another set of current transformers was used but satisfactory results were not obtained. A constant current transformdr core was available in the laboratory. On it were mounted five coils designed to stand 5000 volts when in series. The core has three legs. It was planned to use one coil on each leg and the whole as. a three phase transformer. The secondary being formed by winding two turns of the cable around each leg. This scheme was tried, about 1000 volts being applied on one coil of the primary. Enough current passedtthe secondary line. The 1000 volts were obtained through a step up single phase transformer belong- ing to the laboratory. Two other similar transformers were necessary. Furthermore, they must be designed to carry 10 amps or their high volt- age sides, as it takes 10 amps to energize the three phase coil and give the required current in the line. No transformers were found which could meet the requirenents. It was then found necessary to build a special transformer. The three legged core was used. Forty-five turns of No. 13 cable on each leg constituted the primary coils of the transformer and two turns of the line cable the secondary coils. The connection was made a Wye-Wye. To change the current in the line the applied voltage was changed by changing the field of the supply alternator in the laboratory. The speed was always kept constant giving a constant frequency of 60 cycles. The current in the line was measured by a step down current trans- former. The free end of the line was short circuited to give the different high currents. It was then necessary to find out a means to measure the voltages induced in the coils. The voltages induced in a one hundred turn coil was estimted not to exceed 1.5 volts. Low LC. voltmeters were not available and those found on the market did not satisfy the requirements due to their low resistance. Voltmeters with high resistances allowing but a fraction of an ampere to flow in them was necessary, on account of the back ampere-turns which tend to Oppose the inducing flux. 1.0. galvanometers or vacuum tube voltmeters could have been used if available. All meters were shielded to protect their coils against the direct effect of magnetic lines around the conductors. Furthermore, the leads to the measuring instruments were run perpendicularly to the conductors and for more safety, twisted around each other. An oscillograph was used to measure the induced voltage. The wave on each vibrating element was examined and when found to he sinusoidal was accurately measured. To measure the wave it was included between two boundaries of light projected by the two other elements thus giving twice its maximum value. The distance between the two lines was then measured and recorded. The oscillograph was calibrated at different intervals, using the same leads as those used to record the wave. Thus with the instruments,available in the department it was possible to perform the tests and get accurate results. In regard to the exploring coils, it was first thought possible to use some coils available in the department. When tried it was found that a large voltage was induced in the coils. much larger than was expected. It was discovered that this voltage was largely due to a direct induction from the line to the coils. Two similar coils were then used. One was put around a steel leg and the other outside of it. The two were under similar conditions with respect to the line. They were connected so as to Oppose each other. The oscillograph was then used to register the difference in voltages, viz. the voltage due to the flux in the iron tower. This was found to be small. Due to the fact that at different points on each leg the flux varies, it was decided to build very narrow exploring coils so as to give voltages in a narrow piece of steel. 0n the other band. due to the direct induction from the line, the exploring:coils were wound as thin as possible. In this test the coils were wound with one hundred turns of very thin wire (B. & S. No. 27). They were then put around the legs of the loop and given the same shape, viz.. thewaere bound in the contour of the legs. In this manner the influence of the f1ux.from the lines was reduced to a minimum. By examining the figure it is clearly seen that the fluxes in the three legs were unequal. Further- more. as will be shown . 3 A C later, the flux in c was smaller than either that in a or b. On the other hand. due to the symmetry of the figure, when under balanced load the number of lines in a and b were equal. From another point it was assumed throughout this test that leg a shielded both legs b and c against the magnetic effect of the current flowing in conductor a and that leg b shielded both legs a and c agsinst the current flowing in b. Thus the flux induced in leg c is due to conductor 0 only while that in a and b is due to a and c. and b and c respectively. The necessity for using different exploring coils at diff- erent points along the legs is seen from the fact that the flux was not uniform in any leg. This will be mathematically proven later. Consider one leg a and let us study the nagnetic effect in a due to current flowing in lines a and c. (II is perpendicular and bisects the leg. (cy = 1/2 0Q). AP is another perpendicular fran a. If the effect of c alone is considered, the maximum flux would be at point M, according to the formula I u/‘u 3;; lines per square centimeter r where r is the distance in centimeters. I the current in amperes, and /u the permeability. If the effect of a is conshiered. maximum flux is at P. Due to the fact that both fluxes are aiding and that R is at same distance from a and c the maximum flux would be at R. It is to be noted tint the flux at P0 is largely due to currents in a. At 0 the flux is small. At Q the flux is mostly due to c and is nearly equal to that on the corners of leg c. That point P does not fall on O can be proven as follows: AP - GM, being perpendicular and conductors a and c are at the same distance from Q0. we further have for the same reason cm : CY - AP - aS If P falls at 0 then so - a3. Therefore a0 is perpendicular to Y3 and Q0 or in triangle aOS aS - a0 Angle aSO - aOS . 90° which is impossible. Therefore. P cannot fall on O, viz. the perpendiculars drawn from the outside conductors do not fall at the Joints of the legs. The value of z in terms of r is found as follows: QM - r tan 60 . r v3- l.73205 r “R - r tan 30 ”TI 3 0.57735 r M? - 21MB 1.154? r QP - 1.75205 r + 1.547 r r [2.88675] N I v(2.88675 r)2 + 52 r [V‘e'.3'3"3""] 3.05504 r To find g in terms of r 2P0 n GMT-'MP 1.75205 r - 1.1547 r - .57735 r .1 g - r + 0. r 2 a 1.1547 r w-cr-W 3 ”TI? 3 - 1.5275 r CR afiRa - g Cos 30 . 1.1547 Consider now this other fhgure. The c: flux at any point along e9” X! can be found in ' r terms of the flux at x x x m and the angle 9. The flux at x is Atmitis Ems/u.21 Cm ‘PQZICOSQ 10 r s Bx Cos 9 Theoretical solution: Assume a conductor (“L carrying a current I abamperes r < l flowing as diown in the figure. IL \ It is required to know the 6 /7> 4 f 7‘! magnetic intensity at a point P outside of that conductor. To solve this problem consider an elemmt of length dx at distance L centimeters from P. By Coulomb's law the effect of that element on a point 1. centimeters away is dHI-Ji%_x_ 0036 Therefore the effect of the total wire is H-ffJigpaidx assuming the conductor to be very long. Fran the figure we have x-srtane Therefore dx s r secz 6 d 0 Also I. . g Cos e Inserting these values in the above expression 1! III-f? Lgoserseczgdo -w r72 2 Cosz 6 11 11 This becomes H - f2 I gos g d 6 .1! r '2 r a 2 I r If I is in amperes H - I2 I gausses r If the point P is in a material having /u permeability B . /u iL lines per square centimeter r This is one of the fundamental formula applicable here. In alternating current if the waves are sinusoidal and time properly chosen the current at any instant may be expressed as isImuSinwt where w is the angular velocity w . an and t the time in seconds. The instantaneous current 1 produces a flux Q. Due to the fact that i is varying Q is varying too. In any material which has a con- stant permeability the flux wave has the same shape as the current wave. Due to the fact that the flux density in the tower legs is low it can be assumed that the penneability of the steel within the boundaries used is constant and therefore the flux wave is sinusoidal. Therefore Q - QIn sin w t It was seen that Q. - /u ’2 1 A r where A is cross section area. Therefore Qm . P J? A 12 and Qs/u,2]nsinwta r In a balanced three phase system the currents are 120° apart. 0 Assuming the currents to be i; . In sin w t 1b" 1m sin (wt + 12.0) 1,-- 1m sin (wt + 240) Therefore, the general fomulae for the fluxes due to the three currents are: Qficotnsinwtn/u.2h Asinwt 1' Qb-Qmsin(wt+120)-/u.21m Asin(wt+120) r and Qc-Qmain(wt+24o)-/u.21mlsin(wt+24o) r First, to find the fluxes at different points on leg 9. viz. at Q. M. B. P and O. . Qb has no effect on leg a because leg b is acting as a shield to log 0. against current in conductor b. The flux at 0 due to current a is QM-OAsinwt 1.154? r where C - /n A The flux at 0 due to C is Q6” - 0% sin (wt + 240) Therefore the total flux at o is cit-o " Qa-o " Qc-e Qt —° 13 -C.§L. Isinwt - sinjwt+3§gQ11 r 1.1547 2 Similarly the fluxes at P. R. M and Q are respectively Qt-P - C I1?. In [ sin wt - siant + 249)} r 1.5275 Chas-0,213, [sinwt - sin (wt+240)] 1.1547 r Qt'ii - 0 I2 In: ‘ sin wt - sin (wt + 240) r 1.5275 1 ' a C 2 sin - sin w ' + 2 Qt4g 4.;1m_ E____JEL. ._____LJL___&Ql] 3.05504 0n the other hand Therefore °o . . . where K 3 e0 . sun-H dg 10-6 dt -N0 :2 La 10"6 d sin wt - sin (1g + 24m] r dt 1.1547 2 -NC 2 1m 10"9 [w cos wt - w Cgs (wt + 2491} r [ 1.1547 2 «K'E cos wt - Cos (wt + 24911 1.1547 no 2 g... 10-9w r -K [ cgs wt - 903 wt figs 259 - sin fl sin 249! 1.1547 2 --K [.866 Cos wt + .25 Cos wt - .433 sin wt] -K [1.116 Cos wt - .433 sin wt] It! maximum value occurs when -1.116 sin wt s .433 Cos wt and minimum when Its minimum is then 60m. wt - 158° 45' wt - ~21° 15' -K [1.116 x .93201 + (.433 x .55244)] 14 e e -K [1.04012 + .156956] - -l.l9705 K and its maximum °am - -K [1.116 x (-.93201) - (.455 x .36244” - + 1.19705 K Similarly e1, . ~K [Cos wt - Cos (wt + 2491] 1.5275 - -K [Cos wt + .327233 Cos wt - .56693 sin wt] - -K [1.327 Cos wt - .566 sin wt] Its minimum occurs when tan wt _- -.42652 wt - -23° 5' Its maximum when wt :- 156° 54' It is then +K [1.327 x .91982 + .566 x .39234] K [1.2206 + .22206] 1.4426 K e. I- -K [.866 Cos wt + .433 Cos wt -r .75 sin wt] -K [1.299 Cos wt <- .75 sin wt] Its maximum value occurs when tan wt '- -.577 wt .. 150 It is then em a- 1: [1.299 x .86603 + .75 x .5] a- [1.1249 + .575] K ‘105K 15 ’ u-KECgsfl - Cos (wt+24o) 9“ 1 5275 J 3 -K [.6546 Cos wt + .5 Cos wt - .866 sin wt] - -K [1.1546 Cos wt - .866 sin wt] Its maximum is when tan wt . .75 wt - 145° 7' It is then 911171 - K [.5196 + .9235] x [1.4451] -K Efifi - O s z + 2 i -K [.5275 Cos wt + .25 Cos wt - .455 sin wt ] °Q -K [.5775 Cos wt - .455 sin wt ] It has its maximum value when tan wt - .75 wt - 145° 7' It is then m - K [.5773 x .79986 + .433 x .60019] I- K [.4617 + .2598] - .7215 K Leg b Q. has no effect on leg "b". The flux at 0' due to current b is aka-c .21., sin (wt+1z.o) 1.1547r The flux at 0" due to c is Q”, .. c 1.: In sin (wt + 240) 0’ 1‘ 16 Therefa'e the total flux at 0' is Qt-o' ' Qa-o' - Qc-o' Qt...” " 043.1111... [81niwt + 1201 - __s;i_n [wt + 240) r 1.154? ‘ 2 Similarly the flux is at P', R3. M' and Q are respectively Q._P. - 0._i§_hn; [sin (wt + 120) - sin (Wt + 2401] r 1.5275 Q..R. - c .2 Im [sin (wt + 120) - sin (wt + 240)] r 1.1547 Qt-M' - C .2 In [sin (wt + 129} - sin (wt + 240)] r 1.5275 G .2 1m (sip (gt + 129! - sin (33 + 2491] 3.05504 2 i 4... Consequently so. = -'N0 .2 1m 10"6 d [sin (wt + 1291 - sin (wt + 240)] r dt L 1.1547 ' 2 -NC 2 1.. 10.9 wlcgs (wt + 1291 - cos (wt + 2491] r 1.1547 2 -K [.866 cos(wt + 120) - .5 cos (wt + 240)] -K [.866 (-.5 cos wt - .866 sin wt) - .5 (-.5 cos wt + .555 sin wt)] K [.433 cos wt +.75 sin wt - .25 cos wt + .433 sin wt] K [.183 008 wt + 1.183 sin wt] It has its maximum when .183 sin wt a 1.183 cos wm tan‘Wt 7.13M51. .183 a 6.464 81° 12.5' 3- 17 It is then 820' . K [.185 x .15281 + 1.183 x .98825] c K [.02796 + 1.16909] = K [1.19705 Similarly EP' - éK [cos (wt + 120)._.ggs [th+ 240)] 1.5275 - K {.5 cos wt + .866 sin wt - .5 cos wt - .866 sin wt] 1.5275 - K [.5 cos wt +.866 sin wt - .327233 cos wt + .56693 sin wt] 3 K [172767 COS‘Wt + 1.33293 sin wt] It has its maximum when .173 sin wt - 1.3332 cos wt tan wt - 7.7151 wt . 82° 55'.4 It is then EMP' 8 K [.173 x .12870 + 1.3329 1 .99168] . K [.0222551 + 1.52191] -. K [1.344175] BR. = -K [.866 cos(wt + 120) + .433 cos wt - .75 sin wt)] 4K [.866 (-.5 cos wt _ .866 sin wt) + .433 cos wt - .75 sin wt] K [.433 cos wt + .75 sin wt - .433 cos wt + .75 sin wt] s x [1.5 sin wt] This is maximum when sin wt 8 1 wt - 90 It is thm ELIE. ‘ 1.5 K e ,n —K cos wt t_1201 - cos_Lwt + 2401] ‘45 1.5275 18 - K [.6546 (.5 cos wt + .866 sin wt) - .5 cos wt + .866 sin wt] . K [.3273 cos wt + .5668 sin wt - .5 cos wt + .866 sin wt] = K [1.4328 sin wt - .1727 cos wt] This is maximum when 1.4328 cos wt + .1727 sin wt = 0 tan wt s —8.2964 wt . 95° 52.41 It is then °mn' = K [1.4328 x .99281 + .1727 x .11968] 3 K [1.42249 + .02066] - K [1.44315] E“ = 6K [cos (wt + 120)._ cos (wt :fi240)] L 3.05504 2 K [-.08635 cos wt + .71644 sin wt] This is maximum when .08635 sin wt + .71644 cos wt s 0 01‘ tan wt 3 ‘8.296 wt a 95° 52.4' Its value is then . K'[.010334 + .711288] = K [.721522] K [.3273 (.5 cos wt + .866 sin wt) + .5 (-.5 cos wt + .866 sin wt)] K [.16365 cos wt + .28344 sin wt - .25 COS'Wt + .433 sin wt] 19 Le c g 476' The flux.at F is Q '- C .2 In sin (wt + 240) r 0—? 1' ' 'F’ Therefore the voltage at F is 0 D E ec-F -tNC .2 Im w cos (wt + 240) r - -K cos (wt + 240) - éK [-.5 cos wt + .866 sin wt] - K [.5 cos wt - .866 sin wt] It has its maximum value when --.5 sin wt - .866 cos wt 3 0 tan wt - -l.732 we s 300 It is then em - K [.5 x .5 - (-.866 x .866)] - K Fo = r 7'5” . 1.732 r EB - .866 r 03 ' Vrz + (.866 r)2 "u— EE' 7 . 1: (2,5457) a 1.3228 r 2 ec—E -‘K [.5 cos wt - .866 sin wt] L 1.3228 1.5228 20 e K [.3779 cos wt - .6546 sin wt] It has its maximum when wt a 300 It is then K [.3779 x_.5 + (.6546 x .866)] cE K [.18895 + .56688] .75583 K 80 a K [.5 20s wt - .866 sin wt] K [.25 cos wt - .433 sin wt] It has its maximum when wt .. 300 It is then K [.25 x .50 + .433 x .866] 0 ll K [.125 + .385] K [.50] .2235 K It is to be noticed that there is a phase angle between the different voltages at different points on the lower halves of legs a and b. This difference in the phase angle on the sane leg is very small, being only 15° on legs a and b. The upper halves have the same angle or a very small difference in phase angle. This constancy in the angles shows that the flux in the upper halves is almost a pulsating plux. viz. it is largely the to the current in conductor c. Leg c has no difference in its voltage phase angles at different points. The voltages along c are in phase. On the other hand, the voltage is pr0portional to the flux, thus 21 cam-=27 Therefore, the fluxes at different points is proportional to their voltages. The fluxes on leg a at o, P. R. M and Q are respectively Qmo - D [1.19705 K] 0m, - D [1.4426 K] om - D [1.5 x] 0m - D [1.4451 K] QmQ - D [.7215 K] where D is a constant depending upon frequency and number of turns. Similarly on leg b the fluxes at 0'. P'. R'. 11", and Q' are respectively D [1.19705 K] 5” D [1.34475 K] 5.“ D [1.5 K] 5: D [1.44315 K] s: D [.7216 I] if The average flux on a or b is Q. - pg [1.19705 + 1.4425 + 1.5 + 1.4451 + .7215] . 5 [1.26085] 0n leg c Qmo - D [.2235 K] QmE - D [.75583K] om, - D [1.000 :3 Q1111) - D [.75583 K] Qmo' a D [.2235 K] Qm s DK [2.95866] 3 - D [.59175 x] 22 Therefore the flux in each of the legs a and b is about 1.26985 times the flux in c .59173 01' Q8 ’ Qb . 2013 Q0 The hysteresis losses in a and b are (2.13))"6 times those in c. . 1.6 or PM - rhb - (2.15) rho - 3.3528 Phc 2 The eddy current losses in a and b are each (2.13) times those in c P P (2 13)2 P as 6b ' so If all the losses were due to hysteresis alone, the total losses in each of the legs a and b would have been P - P - 3.3528 P ta ‘ tb hc viz. the losses in c would then be only 1 of the total losses or 7.7056 about 136’. If all the losses were due to eddy current alone. the total losses in each leg (a and b) would have been P a - Ptb . 4.5369 Ptc t viz. the losses in c would then.be l of the total losses or about 10.0738 10%. The actual losses in c are between those two values. viz. between 10 and 13% while from 43 to 45% of the total will be the losses in each of the legs a and b. Leg c was not shielded by any line. Under present conditions the values of voltages at different points on leg c are different from what 23 was found previously. To calculate these voltages procede as follows: A0 - 1.1547 r So . .57735 r A: c :a E0 - .866 1 : I I a 00 - 2 r I 1 : L-ii - .7 h--_l cs - 1.5225 r 0' SD F E 0 3 Therefore AXE-310+ . 102 r V1 + 2.082245 1.755 r. Viz + (.57735 + 1.732)2 r2 E; r 7 l + 5.331481 2.516 r. AD - r vr‘I—(.57735 + 2.595)2 r v1 + 10.050525 3.33 r. 10' r V1 + (.57735 + 3.454); r v1 + 16.329681 4.163 r. Q - 0 [sin wt sin (wt + 129) + sin.(!§ + 249)! 0 1.1547 4.163 2 QB a G lain wt + sin (wt + 129) + sin (wt + 249)} 1.775 3.33 1.3228 + QT . c Esin wt + sin (wt + 129) + sin (wt + 240)] 2.516 2.516 QD ’ C E§$£_EE_+ sin (wt + 129) + sin (wt + 240)] 3.33 1.755 1.3228 Qoo- 0 [sin wt + sin (wt + 120) + sin (wt + 240)] 4.153 1.1547 2 J Therefore 60‘ 24 -K loos wt + cos (wt + 120) + cos.(wt + 249)] 1 1547 4.153 2 1 4K [.866 cos wt t .2402 (cos wt + 120) + .5 (cos wt + 240)] éK [.866 cos wt - .2403 (.5 cos wt + .866 sin wt) - .5 , (.5 cos wt - .866 sin wt)] éK [.866 cos wt - .1201 cos wt - .2080 sin wt - .25 cos wt + .433 sin wt] -K [.4959 cos wt + .225 sin wt] It has its maximum value when It is then 6on .4959 sin wt = .225 cos wt tan wt - .45372 wt . 204° 24.51 - K [.4959 x .9105 + .225 x .41325] E [.45165 + .092953 .54463 K. -K cos wt + cos (wt3 + 3129) + cgs (wt + 249)] E1. 755 1.3228 éK [.5595 cos wt + .3 cos (wt + 120) + .7559 cos (wt + 240)] 4K [.5698 003 wt + .3 (-.5 cos wt - .866 sin wt) + .7559 (-.5 cos wt + .866 sin wt)] éK [.5698 cos wt - .15 cos wt - .2598 sin wt - .37795 cos wt + .65461 sin wt] - 4K [.0419 cos wt + .3948 sin wt] This is maximum when It is than .0419 sin wt : .3948 cos wt tan wt - 9.42243 wt . 253° 55.51 25 53 - K [.0419 x .1055 + .2948 x .99442] :K [.00442 + .39259] B .39701 K e a -K (cos wt + cos (wt + 129) + cos (wt + 240)) F 2.516 2.516 - -K [.3974 cos wt + .3974 0-.5 cos wt - .855 sin wt)4(.5 cos wt ~ .866 sin wt.} - an [.3974 cos wt - .1987 cos wt - .344148 sin wt - .5 cos wt + .866 sin wt] K [.3013 cos wt - .522 sin wt] This is maximum when -.3013 sin wt - .522 cos wt tan wt = - .522 .3013 wt a 300 It is then eFm . K [.3013 x .5 + .522 x .866] K [.15055 + .452052] . K [.500702] . .500702 K eD a éK (008 wt + cos (wt + 129) + cgs (wt + 249)) 3.33 1.755 1.3228 - éK [.3 cos wt + .5698 cos (wt + 120) + .755 cos (wt + 240)] I éK [.3 cos wt + .5698 (-.5 cos wt - .866 sin wt ) + .7559 (-.5 cos wt + .866 sin wt)] 2 -K [.3 cos wt - .2849 cos wt - .4934 sin wt - .37795 cos wt + .65461 sin wt] K [.3629 cos wt - .1612 sin wt} It has its maximum value when -.3629 sin wt 3 .1612 cos wt 26 ton wt = «.4442 wt .. 335° 3' It is then sqxal to 9m - K [.3529 x .9139 + .1512 x .40594} K [.33155 + .07543] - .40708 K 90' . ~K [cos wt + cos_Lwt + 120) + cos_(wt + 240)) L4.153 1.1547 2 . éK [.2402 cos wt + .866 cos (wt + 120) + .5 cos (wt + 240)] s éK [.2402 cos wt + .866 (-.5 cos wt - .866 sin wt) + .5 (-.5 008 wt + .866 sin wt)] . éK [.2402 cos wt - .433 cos wt - .75 sin wt - .25 cos wt + .433 sin wt] K [.4428 cos wt + .317 sin wt] It is maximum when .4428 sin wt . .317 cos um tan wt . .7158 wt - 35° 35.51 It is than equal to 601! -= K [.4428 x .81315 c .317 x .58200] - X [.36006 + .18449] . . 54455 K We thus see that under these conditions we have a rotating flux in leg c. The voltages at different points of the legs reach their Inaxima at different times. This was found to be true experimentally. Reasons why losses in legs a and b are unequal. Although due to summetry the fluxes in legs a and b should be 27 equal, in the test they were found to be unequal. This was due to the fact that the currents were not 120° apart as assumed in the theory. The impedances of the three current transformers were unequal in value and the fluxes in the potential transformer were not exactly 120° apart. To prove this last statement assume 11, 12, and 13 to be P P the three instantaneous ""¢ -' ’ 9’. magnetizing currents, Q1, Q2, and Q3 the instantaneous 7? I? z fluxes, a the reluctance of 2 7? each one of the outside legs, (f; )0; 9? J R that of the inside log, P and B the reluctances of each part _, 4’ ‘fi' 71’ of the yoke as shown in the figure. Therefore .4-1rn (11 - 12) - (ZR + R + P) Q1 ~RQ2 .41rn(13 ~12) - (28 +11 + P) 83 ”3‘22 and Q1 + Q2 + Q3 It 0 By adding .417 n (11 + 5.3 - 212)-(ZR + B + PHQI + H3) - ZRQZ and knowing that 11 + 12 + 13 e 0 therefore 11 + 12 t -13 and by substitution we get .41: n (4512) = (a + a + P) (~52) - mg .4511 (312) . 02(411 +5 + P) (1) (2) I’“ 28 Therefore .477 n 12 - 121 +33 + P Q2 (3) Equation (3) shows that the flux Q2 18 in phase with the magnetiz- ing current 12 in the middle leg. Substituting (3) in (l) we get .4wni ~g+g+20 =(2R+R+P)s -RQ 1 3 2 1 2 which becomes .4nn11-(m+a+r)e1+n_+_mfit2 (4) 3 This clearly shows t'rat the magnetizing current is not in phase with the flux in leg 1. By substituting (3) in (2) we similarly get .4wni3-(3l+R+P)Q3+B_+_§_t£ Q2 which means the same thing as for mgnetizing current 11. The flux in leg 1 though is lagging the current while that in leg 0 is leading. When the transformer is fully loaded, viz. when the total current input is large compred to the magnetizing current the flux in each leg becomes more in phase with the magnetizing current. In the case under consideration the load current input in the primary never was more than ten times the mgnetizing current. There- fore the secondary voltages never were 120" apart. To this add the effect of the impedances which produced another phase difference. _ has the currents were not 120° apart and the fluxes induced in legs a and b were not the me. 29 Procedure to obtain results from the oscillographic readings: The oscillograph gave the height of the naximum voltage. The effective value was then computed. The resistance of the osci llographic elanent and the outside resistance in series with it was measured. Fran the voltage applied on the oscillograph and the resistance of the element and the rheostat in series with it the current flowing in the oscillograph and the exploring coil was computed. The impedance of the coil was next measured. Its direct current resistance was taken. From these two values the reactance of the coil was computed. The drop in voltage in the oscillographic circuit is ohmic. To the value of the voltage across the oscillograph was directly added the ir dr0p in the coil and vectorially the ix drop. The total voltage gave the induced voltage in the coil. Knowing the induced voltage and referring back to the formula for the induced voltage in a transformer the mximum flux was found. Know- ing the cross sectional area of the legs the maximum flux density was computed and thence the hysteresis and eddy losses. To get the average losses the average flux was used. It is to be noticed that losses-distance curves were plotted for one set of readings. The areas under these curves were measured by a planimeter ani the average loss taken. This checked very closely with the results of average losses found by using the average flux. Formulae used on computation: a. The impedance of the coil is found by ZBE I 30 where Z is impedance in ohms E the effective voltage (alternating current) I the effective current ( " " ) The direct current resistance is R-E I where R is resistance in ohms E direct current voltage in volts I direct currait in amperes The reactance of the coil is the: X " m2 ohms. (Many readings for E3115, 1‘“, Edir' and Idir were taken. The average values of Z and B were taken to compute X.) b. The resistance of the element was found by applying direct current and taking readings of current and voltage. c. The distance measured on the oscillograph is twice the maximum value. Half of that distance gives the maximum value. The effective voltage is Eeff " ”07 no where Em denotes the mximum value. The total induced voltage is 1at " ‘7: + ir)‘3 + (if)5 31 where e is the load voltage in phase wdth current i the current flowing in the circuit ir the drop across the ohmic resistance of the coil r the ohmic resistance of coil ix dr0p across inductance of’coil x reactance of coil. (1. The mimum can be calculated by the following fonnnla ce Eeff - 4.44 If f Q10 10 volts Therefore on - n x 19° lines where gm is maximum flux E the effective induced voltage N number of turns in 0011 f frequency in cycles per second. The flux density is found thus 3m ‘._Sha. .A where A.is area of cross section in square centimeters e. The hysteresis loss is computed by means of’Steinmetz' empirical fonnula f 1'6 1'7 tt tim ter Pb - Eh an 0 we 3 per square con 8 where K5 is coefficient of hysteresis f. The eddy loss is found thus Pe - ”2 h2 f2 B: 10-15 watts per square centimeter 6.P where P is resistivity the material h thickness of the sheet 3. The total losses are the sum of the hysteresis and ed current osses Pt 3 Ph + p9 32 Results and Computation: a. Resistance and impedance of coil At 60 cycle frequency the impedance of the exploring coil was found to be Z c 7.556 ohms Its direct current resistance was R,- 7.22 ohms Its reactance is then x - v (7.555)? - v(7.22)‘°‘ - 2.24 ohms b. Resistance of the element and outside rheostat in series. This was found to be I! - 9.5 0. Applied load voltage. The toltage applied from theczoil on the oscillograph gave a deflection of 3/8'(lst set of readings on leg a with 360 amp.) With a direct current voltage of 2 volts the deflection was 9/8”. The maximum value of voltage was then 'Em - 2 x 8 x 3 I .333 9 x 8 x 2 Its effective value was then 9 I- .235 7011; 8 The current flowing in the oscillograph and the coil was therefore I s .235 9.5 ’ .0248 map 0 33 The ir dr0p in the resistance of the coil was ir - .0248 x 7.22 - .179 volts The ix drop was ix = .0248 x 2.24. r .0556 The total induced voltage was Et " V (.235 + .Ws + . 5')‘2 ' V .171396 + .003091 ' V 174487 = .417 d. Flux The maximum flux was therefore Q... It .417 x 198 4.44 x 50 x 100 c 1565 lines The area of the cross section being 3/8” or 2.418 sq. cm.. the maximum flux density was Isl-.1251 m 2.418 - 648 lines per square centimeter e. Hysteresis losses The coefficient of hysteresis for that sample of steel is 35“ .015 The hysteresis loss was therefore 1.6 .7 Ph ' .015 x 60 x 648 x 10 watts per sq. cm. - 28.36 x 10‘4 watts per sq. cm. f. 8. 34 The eddy current loss The resistivity of steel at ordinary temperature is P c 19 microhm centimeter The eddy cirrent coefficient is therefore - (3.1415.x lo6 1 (.315)2 6 x 19 = 8.61 x 103 Ks .315 is the thickness of the steel in centimeter being 1/@ of an inch. The eddy current loss is therefore P6 - 8.61 x 10? 1 (so)2 x (648)2 x 10"16 s 15.0 x 10-4 watts per sq. an. The total loss per square centimeter is pt . (28.36 + 13) 10‘4 3 41.36 x 10-4 watts per sq. cm. Measurement of the Impedance of 0011 Frequency 60 cycle. Direct Current I I z ohl I .58 7.57 7.57 .4 .65 4.75 7.54 .66 .69 5.22 7.56 .5 .8 6.05 7.56 .3 .88 6.65 7.55 .23 Average 7 .556 x I 2.24 Resistance of elenmt and resistance 1 V’ B.ohm .1 .95 9.5 .2 1.94 9.7 .31 2.9 9.35 .596 3.75 9.46 .5 4.73 9.46 Average 9.5 2.92 4.8 3.6 2.15 1.65 Rohm 7.3 7.27 7.2 7.16 7.17 7.22 55 Direct Current Resistance of Coil Y A Bee Coil Average 2.92 .4 7.3 ) 4.6 .66 7.27; 5.6 .5 7.2 ; Coil 7.22 2.15 .3 7.16: ) 1.55 .23 7.17) .95 .1 9.5 ) 1.94 .2 9.7 g 2.9 .31 9.55; Set 9.5 3.75 .396 9.46; 4.73 .5 9.46: 560 510 230 1b 888 310 230 I. 1 2 11m 5 888 8/8 11/16 18/18 1 8/4 18/18 1/2 I. - 19.5 Res on 9 810 5/18 9/16 11/18 9/16 Res on 9 288 1/4 7/16 8/16 1/2 Res on 9 440 7/16 18/18 1 19/16 Bee on 9 Phase Angle a - b . 60° a-c-30° b 2 3 {-60 5/8 7/16 1 - 60 1/2 5/18 : - 60 1 1/2 1.60 It was noticed that there was a phase difference in the fluxes at different points on c. which amounts to around 600 between 1 and 2. 2 and 3. There was a very little angle between 2 and 3 on a and 1 and 3 on b. but no angle between 1 and 2 on a and 2 and 3 on b. 37 1 2 3 5/16 8/8 1/4 1/4 5/16 8/18 8/18 1/4 8/18 5/18 7/16 1/4 .333 .611 .722 .277 .611 .222 .388 .472 .388 .722 .888 Off. .431 .510 .195 .353 .157 .274 .274 .510 .628 .666 .722 .444 .527 .555 .388 .444 .444 .277 1.054 .888 eff. .470 .510 .314 .372 .392 .274 .314 .314 .195 .744 .628 .314 .277 .888 .222 .222 .277 .166 .166 .222 .166 .277 .388 .222 eff. .195 .157 .157 .195 .117 .117 .157 .117 .195 .274 .157 360 310 230 440 360 310 230 440 360 310 230 440 38 60 6O 6O 6O .235 .431 .510 .195 .858 .157 .274 .333 .274 .510 .628 I .0248 .0454 .0537 .0205 .0372 .0454 .0165 .0288 .0351 .0288 .0537 .0661 Total e - r - 7.22 x - 2.24 Ir (e+1r) .179 .327 .388 .148 .268 .327 .119 .208 .253 .208 g 533 .477 +1? .414 .758 .898 .343 .621 .758 .276 .482 .586 1.105 ‘0' X BETA IX .0556 .101 .120 .0459 .0834 .101 .0369 .0627 .0787 .0645 .120 .148 (1x)z .003091 .010010 .014400 .002106 .006955 .010010 .0018818 .0089812. .0061436 .0041602 .014400 (o')2 2 (8+Ir) .171396 .574564 .806404 .117649 .385641 .574564 .076176 .232324 .343396 .232324 .806404 e' z+IXz .174487 .584574 .820804 .119756 .392597 .584574 .077538 .236255 .349590 .236484 .820804 °t .417 .764 .906 .346 .626 .764 .278 .486 .591 .486 .906 .021904 1.221025 1.242929 1.114 e . e.m.f. recorded by oscillograph re - resistance of element and.outside resistance - 9.5 39 ‘8 1585 2870 3410 1300 2355 2870 1045 1825 2220 1825 3410 4190 in 1565 2870 3410 A1. 2355 2870 Av. 1045 1825 2220 Av. 1825 3410 4190 Ax. s. 648 1190 1410 1080 538 975 1190 901 755 920 702 755 1410 1740 1300 108 En 2.81158 3.07555 3.14922 3.03743 2.73078 2.98900 3.07555 2.95472 2.63649 2.87795 2.96379 2.84634 2.87795 3.14922 3.24055 3.11394 BETA 4.498528 31517 4.92088 83347 5.038752 109450 4.859888 72450 4.369248 4.7824 4.92088 4.72755 4.218384 4.60472 4.742064 4.554144 4.60472 23400 60590 83347 53400 16535 40250 55254 35850 40250 5.038752 109450 10-4‘I hys.1oss 28.36 75.01 98.50 65.20 21.06 54.53 75.01 48.06 14.88 36.22 49.72 32.26 36.22 98.50 5.18588 153060 136.75 4.982324 96020 86.41 83 419904 1416100 1988100 1166400 289444 950625 1416100 811801 187489 570025 846400 492804 570025 1988100 3027600 1690000 Edd.L. “1390 43.9 61.6 36.2 8.99 29.5 43.9 25.05 5.82 17.7 26.25 15.3 17.7 61.6 93.9 52.4 40 Totai Losses w’ ‘w -41536 118.91 160.10 101.4 30.05 84.03 118.91 73.11 20.70 53.92 75.97 47.56 53.92 160.10 230.65 136.81 .470 .510 .314 .372 .392 .274 .314 .314 .195 .744 .628 .314 .0495 .0537 .0331 .0392 .0412 .0288 .0331 .0331 .0205 .0784 .0661 .0331 Ir .357 .388 .239 .283 .298 .208 .239 .239 .148 .565 .477 Ir+e .827 .898 .553 .655 .690 .482 .553 .553 .343 1.309 1.105 .553 DATA Ix .111 .120 .0742 .0879 .0924 .0646 .0742 .0742 .0460 .175 .148 .0742 __2 II .012321 .014400 .005506 .007726 .008537 .004173 .005506 .005506 .002116 .030625 .021904 .005506 2 (o' ) .888929 .806404 .806404 .429025 .476100 .232324 .305809 .305809 .117649 1.713481 1.221025 .305809 -—2 e' +IX .696250 .820804 .436751 .484637 .236497 .311315 .311315 .119765 1.744106 1.242929 .311315 .834 .906 .558 .660 .696 .486 .558 .558 .346 1.320 1.114 .558 41 3350 3410 2100 2480 2618 1825 2100 2100 1300 4960 4190 2100 42 DATA ‘0 10.4 ' 2 Edi. Total Losses 0. Ba log an 1.6 lognm 3m hys.Loss 5m 10 w I!" w 3350 1390 3.14301 5.028816 106875 96.16 1932100 59.9 156.06 3410 1410 3 .14922 5.038752 109450 98 .50 1416100 43 .9 142. 40 2100 ' 870 2.93952 4.703232 50496 45.44 756900 23.5 68.94 Av. 1233 3.0910 4.9456 88200 79.38 1519289 47.2 126.58 2480 1&0 3.01284 4.820644 66152 59.53 1060900 32.9 92.43 2620 1080 3.03342 4.853472 71364 64.22 1166400 36.2 100.42 1825 755 2.87795 4.60472 40250 36.22 570025 17.7 53.92. AV. 955 2.9800 4. 768 58600 52.74 912025 28 . 22 80.96 2100 870 2.93952 4.705232 50496 45.44 756900 23.5 68.94 2100 870 2.93952 4.703232 50496 45.44 756900 23.5 68.94 1300 538 2.73078 4.369248 23400 21 .06 289444 8 .97 30.03 Av. 759 2.8802 4.6m32 40600 36.54 576081 17.87 54.41 4960 2060 3.31387 5.302192 200530 180.47 4243600 131.5 311.97 4190 1740 3.24055 5.18488 153060 137.75 3027600 94.7 232.45. 2100 870 2.93952 4.703232 50496 45.44 756900 23.5 68.94 Av. 1556 2.1920 5.1072 128000 115.2 2420000 75.0 190.20 .195 .235 .157 .157 .195 .117 .117 .157 .117 .195 .274 .157 .0205 .0248 .0165 .0165 .0205 .0123 .0123 .0165 .0123 .0205 .0288 .0165 .148 .179 .119 .119 .148 .088 .088 .119 .088 .148 .208 .119 IR+e .343 .414 .276 .276 .343 .205 .205 .276 .205 .343 .482 .276 IX .0460 .0556 .0370 .0370 .0460 .0275 .0275 .0370 .0275 .0460 .0645 .0370 (11)2 .002116 .008091 .001369 .001369 .002116 .000756 .000756 .001369 .000756 .002116 .004160 .001369 (0')2 .117649 .171396 .076176 .076176 .117649 .042025 .042025 .076176 .042085 .117649 .232324 .076176 2 Induced E .119765 .174487 .077545 .077545 .119765 .042781 .042781 .077545 .042781 .119765 .236484 .077545 'E .346 .417 .278 .278 .346 .206 .278 .206 .346 .278 43 1300 1565 1045 1045 1300 775 775 1045 775 1300 1825 1045 ‘6 1800 1565 1045 Av. 1045 1300 775 AR. 775 1045 775 Av. 1300 1825 1045 AN. 38 538 648 433 539 433 538 320 430 320 433 320 357 538 755 433 575 log 8“ 2.73078 2.81158 2.63649 2.7316 2.63649 2.73078 2.50515 2.6335 2.50515 2.63649 2.50515 2.5527 2.73078 2.87795 2.63649 2.7597 106 1083“ 4.369248 4.498528 4.218384 4.37065 4.218384 4.369248 4.00824 4.2136 4.00824 4.218384 4.00824 4.08432 4.369248 4.60472 4.218384 4.41552 DATA -4‘w 1.6 10 3m hys.Loss 23400 31517 16535 23450 16535 23400 10192 16353 10192 16535 10192 12142 23400 40250 16535 26030 21.06 28.36 14.88 21.10 ” 14.88 21.06 9.17 14.71, 9.17 14.88 9.17 10.92 21206 36.22 14.88 23.42 2 En 289444 419904 187489 290521 187489 289444 102400 18 4900 102400 187489 102400 127449 289444 570025 187489 330625 Ed.L. Tota Losses 10‘4 8.97 13.0 5.82 9.02 5.82 8.97 3.82 5.74 3.82 5.82 3.82 3.96 8.97 17.7 5.82 10.22 10 30.03 41.36 20.30 30.12 20.70 30.03 12.99 20.45 12.99 20.70 12.99 14.88 30.03 53.92 20.70 33.64 44 '1 10-4w 28.36 75.01 98.50 65.20 0 10-4' 13.0 43.9 61.6 36.2 Total 10‘4' 41.36 118.91 160.10 101.4 Average losses per) 1 cu. can. of tower ) 21.06 54.53 75.01 48.06 14.88 36.22 49.72 32.26 36.22 98.50 136.75 86.41 8.99 29.5 43.9 25.05 Eli. 5.82 17.7 26.25 15.3 EYE. 17.7 61.7 93.9 52.4 338. 30.05 84.03 118.91 73.11 38.50 20.7 53.92 75.97 47.56 26.57 53.92 160.10 230.65 136.81 75.01 DATA Summary of Results L - 19.5' b 0 Nb 710 Total Wh we Total 10"“ 10“" 10"" 10"" 104' 10’" 96.16 59.9 156.06 21.06 8.97 30.03 98.50 43.9 142.40 28.36 13.00 41.36 45.44 23.5 68.94 14.88 5.82 20.70 79.38 47.2 126.58 21.10 9.02 30.12 Eye. 55.23 Total 86.04 Edd. 30.81 59.53 32.9 92.43 14.88 5.82 20.70 64.22 36.2 100.42 21.06 8.97 30.03 36.22 17.7 53.92 9.17 3.82 12.99 52.74 28.22 80.96 14.71 5.74 20.45 Edd. 19.67 Total 58.17 45.44 23.5 68.94 9.17 3.82 12.99 45.44 23.5 68.94 14.88 5.82 20.70 21.06 8.97 30.03 9.17 3.82 12.99 36.54 17.87 54.41 10.92 3.96 14.88 Edd. 12.38 Total 38.95 180.47 131.5 311.97 21.06 8.97 30.05 137.75 94.7 232.45 36.22 17.7 53.92 45.44 23.5 68.94 14.88 5.82 20.70 115.2 75.0 190.2 2342 10.22 33.64 Edd. 45.87 Total 120.88 45 1.-360 Ib-371 10:360 Av. 1.4810 1b.821 Ioasos Av. I‘-235 1b-240 10.228 AN. 1.-440 Ib-430 10.440 Av. DATA 1.1.51 2 8 41 2 8 41 2 3 4 470 475 475 5/16 1/2 13/16 7/8 7/8 7/8 11/18 8/8 7/32 7/16 8/8 1/4 5-0-180 L-24" f—|12m§§Q -60 1200 R-9 365 888 885 7/32 8/8 9/16 11/18 11/18 9/16 5/18 8/18 5/18 5/18 3/16 1 . 50 5:1200 818 320 810 8/18 5/16 1/2 9/18 9/18 19/82 1/2 9/82 5/82 9/82 7/82 5/82 3 $1200 .194 .333 .537 .611 .166 .277 .444 .500 eff. .195 .314 .510 .548 .137 .235 .372 .431 .117 .195 .314 .353 max. .776 .776 .611 .333 .611 .611 .500 .277 .500 .527 .250 eff. .548 .548 .431 .235 .431 .431 .353 .195 .353 .372 .314 .177 DATA max. .194 .888 .888 .222 .166 .277 .277 .166 .138 .500 .194 .138 eff. .137 .274 .235 .157 .117 .195 .195 .117 .097 .353 .137 .097 470 365 315 475 363 320 475 365 310 47 f = 60 L 8 24” f - 60 f I 60 .195 .314 .510 .548 .137 .235 .372 .117 .195 .314 .353 .0205 .0331 .0537 .0577 .0144 .0248 .0392 .0454 .0123 .0205 .0531 .0372 .104 .179 .283 .088 .148 .239 .268 a. e+IB .343 .553 .898 .965 .241 .414 .655 .758 .205 .343 .553 .621 DATA IX .0460 .0741 .120. .1291 .0323 .0556 .0879 .101 .0275 .0742 .0834 (1112 .002116 .005505 .014400 .016641 .001043 .003091 .007726 .010010 .000756 .002116 .005506 .006956 (6')2 .117849 .805809 .806404 .931225 .058081 .171396 .429025 .574564 .042025 .117649 .305809 .388641 Induced 32 .119765 .311315 .820804 .947866 .059124 .174487 .436751 .584574 .042781 .119765 .311315 .392597 E .346 .558 .973 .243 .417 .660 .764 .206 .346 .558 .626 48 0n 1300 2100 3410 3660 915 1565 2480 2870 775 1800 2100 2855 1a 1300 2100 3410 3660 Aw. 915 1566 2870 Ar. 775 2100 2355 AN. an 530 870 1410 1520 1084 378 648 1050 1190 811 320 538 870 975 676 108 Ba 2.73078 2.93952 3.14922 3.18184 3.0350 2.57749 2.81158 3.01284 3.07555 2.9090 2.50515 2.73078 2.93952 2.98900, 2.7604 DATA 5 1.6 1.6 logBm 3m 4.369248' 23400 4.703232 50496 6.038752 109450 5.090944 125230 4.8560 71800 4.123984 13305 4.498528 31517 4.820544 66152 4.92088 83347 4.6544 45125 4.00824 10192 4.369248 23400 4.703232 50496 4.7824 60590 4.41664 26100 10“ Eye .1“ 21.06 45.44 98.50 110.90 64.62 11.97 28.36 59.53 75.01 40.51 9.17 21.06 45.44 54.53 23.49 he”; 289444 756900 1988100 2310400 1178000 142884 419904 1060900 1416100 657721 102400 289444 756900 950625 456976 Ed. 10" 8.97 23.5 61.6 71.6 36.5 4.43 13.0 32.9 43.9 20.4 3.82 8.97 23.5 29.5 14.12 49 Total Losses 10-5w 30.05 50.94 160.10 182.50 101.12 16.40 41.36 92.43 118.91 61.01 12.99 30.03 68.94 84.03 37.61 .431 .353 .195 .353 .372 .314 .177 .0577 .0577 .0454 .0248 .0248 .0248 .0372 .0205 .0372 .0392 .0331 .0186 .417 .417 .327 .179 .178 .178 .268 .148 .268 .283 .134 IR+e .965 .965 .758 .414 .609 .609 .621 .621 .655 .553 .311 DATA IX .129 .129 .101 .0556 .0556 . 556 .0834 .0460 .0834 .0879 .0742 .0417 2 (IX) .016641 .016641 .010010 .003091 .003091 .003091 .006956 .002116 .006956 .007726 .005506 .001738 (0')2 .951225 .931225 .574554 .171396 .370881 .370881. .385641 .117649 .385641 .429025 .305809 .096721 2 Induced 0..) .947866 .947866 .584574 .174487 .373972 .373972 .392597 .119765 .392597 .436751 .311315 .098459 'E .973 .973 .764 .417 .611 .611 .626 .346 .626 .660 .558 .313 50 ‘n 3660 3660 2870 1565 2300 2300 2355 1300 2355 2480 2100 1175 5]. DATA b 1.5 10" 2 54.1. Tote. Losses 0. Ba 1038“, 1.6 10gb.ll Bm- Hys.1.. 8m 10' 10 3550 1510 3.17595 5.090944 123230 110.90 2250100 71.5 152.50 5550 1510 3.17595 5.090944 123230 110.90 2250100 71.5 152.50 2870 1190 3.07555 4.92055 53347 75.01 1415100 43.9 115.91 1555 770 2.55549 4.515354 41532 37.37 592900 15.4 55.77 19. 1245 3.0952 4.95232 59500 50.54 .55000 45.1 125.74 2300 950 2.97772 4.754352 55123 52.31 902500 25.0 50.31 2300 950 2.97772 4.754352 55123 52.81 902500 25.0 50.31 2355 975 2.95500 4.7524 50590 54.53 950525 29.5 54.03 1300 535 2.73075 4.359245 23400 21.05 259444 5.97 .30.05 19. 553 2.9309 4.55944 45910 44.01 727509 22.59 55.50 2355 975 2.95900 4.7524 50590 54.53 950525 29.5 54.03 2450 1020 3.01254 4.520544 55152 59.53 1050900 32.9 92.45 2100 570 2.93952 4.703232 50495 45.44 755900 23.5 55.94 1175 455 2.55554 4.295524 19555 17.59 235195 7.33 25.22 Ly. 540 2.9243 4.57555 47730 42.95 705500 21.55 54.5 .137 .274 .235 .157 .117 .195 .195 .117 .093 .353 .137 .098 .0144 .0288 .0248 .0165 .0123 .0205 .0205 .0123 .0103 .0372 .0144 .0103 .104 .208 .179 .119 .088 .148 .148 .083 .0744 .268 .104 .0744 IR+0 .241 .482 .414 .276 .205 .343 .343 .205 .172 .621 .241 .172 DAIA IX .0323 .0645 .0556 .0370 .0275 .0460 .0460 .0275 .0231 .0834 .0231 (111)2 V .001043 .004160 .003091 .001369 .000756 .002116 .002116 .000756 .000533 .006956 .001043 .000533 (0')2 .055051 .232324 .171395 .076176 .042025 .117649 .117649 .042025 .029804 .385641 .058081 .029584 Induce (3) .059124 .235454 .174457 .077545 .042781 .119765 .119765 .042781 .050117 .392597 .059124 .030117 ‘E .243 .486 .417 .278 .206 .346 .346 .206 .173 .626 .243 .173 915 1825 1565 1045 775 1300 1300 775 650 2355 915 650 53 5394 O 1.6 10‘4 10-4 Total Losses .m 5m log 5“ 1.5 1og5m 5m Hys.L. 5: 55.3. 10‘ 915 379 2.57554 4.123954 13305 11.97 143541 4.43 15.40 1525 755 2.57795 4.50472 40250 35.22 570025 17.7 53.92 1555 545 2.51155 4.495525 31517 25.35 419904 13.0 41.35 1045 433 2.53549 4.215354 15535 14.55 157459 5.52 20.70 19. 554 2.7435 4.3595 24525 22.07 305915 9.52 31.57 775 312 2.49415 4.00524 10192 9.17 97344 3.52 12.99 1300 535 2.73075 4.359245 23400 21.05 259444 5.97 30.0: 1300 535 2.73075 4.359245 23400 21.05 259444 5.97 30.03 775 312 2.49415 4.00524 10192 9.17 97344 3.52 12.99 19. 425 2.5254 4.20544 15045 14.44 150525 5.50 20.04 550 259 2.42975 3.5575 7720 5.94 72351 2.24 9.15 2355 975 2.95900 4.7524 50590 54.53 950525 29.45 53.99 915 379 2.57554 4.123954 13305 11.97 143541 4.43 15.40 550 259 2.42975 3.5575 7720 5.94 72351 2.24 9.15 19. 473 2.5749 4.27954 19045 17.14 223729 5.94 24.05 DATA Summary of Results cu. cm. of toner 1 - 24' a b 0 Vi ‘Wc Total W3 ‘W0 Total Wh _ W6 Total 10": 103% 10.477 10". 10.4w 104' 10”: 10.417 10.497 21.06 8.97 30.03 110.90 71.6 182.50 11.97 4.43 16.40 I.-470 45.44 23.5 68.94 110.90 71.6 182.50 36.22 17.7 53.92 Ib-475 98.50 61.6 160.10 75.01 43.9 118.91 28.36 13.0 41.36 Ic-475 110.90 71.6 182.50 37.37 18.4 55.77 14.88 5.82 20.70 64.62 36.5 101.12 80.64 48.1 128.74 22.07 9.52 31.57 Ly. Hys. 49.78 Edd. 31.27 Total 81.15 Ayerage losses per . cu. cm. of tower 11.97 4.43 16.40 52.31 28.0 80.31 9.17 3.82 12.99 1‘9560 28.36 13.0 41.36 52.31 28.0 80.31 21.06 8.97 30.03 Ib-363 59.53 52.9 92.43 54.53 29.5 84.03 21.06 6.97 30.03 1g-550 75.01 43.9 118.91 21.06 8.97 30.03 9.17 3.82 12.99 40.51 20.4 51.01 44.01 22.59 55.5 14.44 5.50 20.04 Av. Hys. 33.02 Edd. 1620 Total 49.22 Ayerage losses per on. am. of toner 9.17 3.82 12.99 54.53 29.5 84.03 6.94 2.24 9.18 I‘-322 21.06 8.97 30.03 59.53 52.9 92.43 54.53 29.46 83.99 13.322 45.44 23.5 68.94 45.44 23.5 68.94 11.97 4.43 16.40 10-322 64.53 29.5 84.03 17.89 7.33 25.22 6.94 2.24 9.18 23.49 14.12 37.61 42.95 21.85 64.8 17.14 6.94 24.08 19. Hys. 27.86 Add. 14.30 Total 42.16 Ayerage losses per 55 DAQA a b o 1. 15 1o 1 2 3 4 5 1 ’ 2 3 4 5 1 2 3 4 5 255 255 255 3/15 7/32 5/15 7/15 3/5 3/5 7/15 7/15 9/32 3/15 5/32 3/15 1/4 5/32 1/5 1 25.5 ‘ up. 1200 Res. 9 ' A f s 60 350 350 350 5/32 1/4 3/5 17/32 7/15 9/15 9/15 1/2 5/15 1/4 5/32 1/4 5/15 7/32 5/32 L - 26.5 sp. 1200 R65. 9 f - 60 445 445 450 1/4 3/5 9/15 3/4 9/15 11/15 3/4 11/15 7/15 5/15 3/1511/32 11/32 1/4 522 L - 25.5 up. 1200 Bee. 9 f - 60 285 350 445 1b I0 255 285 350 350 445 450 L - 25.5 f I 60 5. D$IA .166 .194 .277 .388 .333 .138 .222 .333 .472 .388 .222 1333 .500 .666 .500 off. .117 .137 .195 .274 .235 .097 .157 .235 .333 .274 .157 .235 .353 .470 .353 .333 .388 .388 .250 .166 .500 .500 .444 .277 .222 .611 .666 .611 .388 .277 eff. .274 .274 .177 .117 .353 .353 .314 .195 .157 .431 .470 .431 .274 .195 max. .135 .166 .222 .135 .111 .166 .305 .305 .222 .138 off. .097 .117 .157 .097 .097 .157 .195 .137 .097 .117 .215 .215 .157 .097 56 .117 .137 .195 .274 .235 .098 .157 .235 .333 .274 .157 .353 .470 .353 .0123 .0144 .0205 .0288 .0248 .0103 .0165 .0248 .0351 .0288 .0165 .0248 .0372 .0495 .0372 .088 .104 .148 .208 .179 .0744 .119 .179 .208 .119 .179 .268 .357 .268 e. IB+e .205 .241 .343 .482 .414 .172 .276 .414 .482 .276 .414 .621 .827 .621 DAEA IX .0275 .0323 .0460 .0645 .0556 .0231 .0370 .0556 .0787 .0646 .0370 .0556 .0834 .1110 .0834 (1x12 .000755 .oo1o43 .002115 .004150 .003091 .000533 .001369 .003091 .006194 .004160 .001369 .003091 .006956 .012321 .006956 (5')2 .042025 .058081 .117649 .232324 .171396 .029584 .076176 .171396 .343396 .232324 .076176 .171396 .385641 .683929 .385641 Induced E2 .042781 .059124 .119765 .236484 .174487 .030117 .077545 .174487 .349590 .236484 .077545 .174487 .392597 .696250 .392597 .417 .173 .278 .417 .591 .486 .278 .417 .626 .834 .417 57 775 915 1300 1825 1565 650 1045 1565 2220 1825 1045 1565 2355 3185 1565 58 5451 a _4 Ed.L. Total Losses 1.5 15 5m am 153 an 1.5 1535In 5m Hy.L. 53 15‘4 15". 775 312 2.49415 4.55524 15192 9.17 97344 3.52 12.99 915 379 2.57554 4.123954 13355 11.97 143541 4.43 15.45 1355 535 2.73575 4.359245 23455 21.55 259444 5.97 35.53 1525 755 2.57795 4.55472 45255 35.22 575525 17.7 53.92 1555 545 2.51155 4.495525 31517 25.35 419954 13.5 41.35 Av. 525 2.7215 4.3535 22575 25.3 275575 5.57 25.57 555 259 2.34544 3.5575 7725 5.94 72351 2.24 9.15 1545 433 2.53549 4.215354 15535 14.55 157459 5.52 25.75 1555 545 2.51155 4.495525 31517 25.35 419954 13.5 41.35 2225 925 2.95379 4.742554 55254 49.72 545455 25.25 75.97 1525 755 2.57795 4.55472 45255 35.22 575525 17.7 53.92 1:. 555 2.7515 4.45555 25245 25.41 355525 11.35 35.75 1545 433 2.53549 4.215354 15535 14.55 157459 5.52 25.75 1555 545 2.51155 4.495525 31517 25.35 419954 13.5 41.35 2355 975 2.95955 4.7524 55595 54.53 955525 29.5 54.53 3135 1355 3.11394 4.952354 95515 55.45 1595555 52.4 135.55 1555 545 2.51155 4.495525 31517 25.35 419954 13.5 41.35 Av. 555 2.9531 4.54495 44155 39.73 545555 19.55 59.55 .235 .274 .274 .177 .117 .353 .353 .314 .195 .157 .0248 .0288 .0288 .0186 .0123 .0372 .0372 .0331 .0205 .0165 .0454 .0495 .0454 .0288 .0205 .179 .208 .134 .088 .268 .268 .239 .148 .119 .327 .357 .327 .208 .148 e! IR+e .414 .482 .311 .205 .621 .621 .553 .343 .276 DATA IX .0556 .0645 .0645 .0417 .0275 .0834 .0834 .0742 .0460 .0370 .101 .111 .101 .0645 .0460 (11?2 .003091 .004160 .004160 .001738 .000756 .006956 .006956 .005506 .002116 .001369 .010010 .012321 .010010 .004160 .002116 (5'12 .171396 .232324 .232324 .096721 .042025 .385641 .385641 .305809 .117649 .076176 .574564 .683929 .574564 .232324 .117649 Induced 32 .174487 .236484 .236489 .098459 .042781 .392597 .392597 .311315 .119765 .077545 .584574 .696250 .584574 .236484 .119765 E .417 .486 .313 .206 .626 .626 .558 .346 .278 .764 .834 .764 .346 59 9m 1565 1825 1825 1175 775 2355 2355 2100 1300 1545 ' 2875 3135 2875 1825 1300 60 5551 b 1.6 10_4 2 Ed.L. Total Losses on 3m log an 1.5 1og5In 5m Hy. L. 5m 15'4 15‘4w 1555 545 2.51155 4.495525 31517 25.35 419954 13.5 41.35 1525 755 2.57795 4.55472 45255 35.22 575525 17.7 53.92 1525 755 2.57795 4.55472 45255 35.22 235195 17.7 53.92 1175 455 2.55554 4.295554 19555' 17.59 235195 7.33 25.22 775 325 2.49415 4.55524 15192 9.17 152455 3.52 12.99 Av. ' 593 2.7731 4.43595 27345 24.55 351549 15.9 35.55 2355 975 2.95955 4.7524 . 55595 54.53 955525 29.5 54.55 2355 975 2.95955 4.7524 55595 54.53 955525 29.5 54.55 2155 575 2.93952 4.953232 55495 45.44 755955 23.5 55.94 1355 535 2.73575 4.359245 23455 21.55 259444 5.97 35.53 1545 433 2.53549 4.215354 15535 14.55 157459 5.52 25.75 15. 755 2.5797 4.55752 45555 35.45 574554 17.5 54.25 2575 1195 3.57555 4.92555 53347 75.51 1415155 43.9 115.91 3135 1355 3.55549 4.952354 95515 55.45 1595555 52.4 135.55 2575 1195 3.57555 4.92555 53347 75.51 1415155 43.9 115.91 1525 755 2.57795 4.55472 45255 35.22 575525 17.7 53.92 1355 535 2.73575 4.359245 23455 21.55 259444 5.97 35.53 Av. 994 2.9974 4.79554 52555 55.25 955535 35.55 55.95 .098 .117 .157 .098 .078 .098 .157 .195 .137 .098 .0103 .0123 .0165 .0103 .0082 .0103 .0165 .0205 .0144 .0103 .0123 .0226 .0226 .0165 .0103 .0744 .088 .119 .0744 .0592 .5744 .119‘ .145 .154 .0744 .088 .155 .155 .119 .0744 e+IB .172 .205 .276 .172 .137 .172 .276 .343 .241 .172 .205 .370 .370 .276 .172 DATA IX .0231 .0275 .0370 .0231 .0183 .0231 .0370 .0460 .0323 .0231 .0275 .0506 .0506 .0370 .0231 (1372 ‘.000533 .000756 .001369 .000533 .000334 .000533 .001369 .002116 .001043 .000533 .000756 .002560 .002560 .001369 .000533 (e'f‘ .029584 .042025 .076176 .029584 .018769 .029584 .076176 .117649 .058081 .029584 .042025 .136900 .136900 .076176 .029584 E2 .050117 .042781 .077545 .030117 .019103 .030117 .077545 .119765 .059124 .030117 .042781 .139460 .39460 .077545 .030117 Induced E .173 .206 .278 .173 .139 .173 .278 .346 .243 .173 .373 .373 .278 .173 61 °m 555 775 1045 650 523 .650 1045 1300 915 650 775 1400 1400 1045 650 .5135: 555 259 775 312 1045 433 650 269 523 216 Av. 500 650 269 1045 433 1300 538 915 378 650 269 Av. 377 775 312 1400 580 1400 580 1045 433 650 269 Aw. 435 log Bm 2.42975 2.49415 2.63649 2.42975 2.33445 2.4771 2.42975 2.63649 2.73078 2.57864 2.42975 2.5763 2.49415 2.76343 2.76343 2.63649 2.42975 2.6385 DATA 1.5 log 5. Bi' 3.8876 4.00824 4.218384 3.8876 3.73512 3.96336 3.8876 4.218384 4.369248 4.123984 3.8876 4.12208 4.00824 4.421488 4.421488 4.218384 3.8876 4.2216 7720 10192 16535 7720 5434 9190 7720 16535 23400 13305 7720 13243 10192 .25392 26392 16535 7720 16656 15“ Hy.L. 6.94 9.17 14.88 6.94 4.89 8.27 6.94 14.88 21.06 11.97 6394 11.91 9.17 23.75 23.75 14.88 6.94 14.99 33. 72361 97344 187489 72361 46656 90000 72361 187489 289444 143641 72361 142129 97344 336400 336400 187489 72361 189225 15“ 2.24 3.52 5.52 2.24 3.82 10.04 10.04 5.82 2.24 5.87 Ed.L. Total losses 10-4w 9.18 12.99 20.70 9.18 6.38 11.06 9.18 20.70 30.03 16.40 9.18 16.32 12.99 33.79 33.79 20.70 9.18 .20.86 DATA Sumary of Results L - 26.5“ a b c wh we 77. wh V. wt W11 we wt 15".. 15“. 15“. 15". 10.477 15'4w 15‘4w 1o". 15% 9.17 3.82 12.99 28.36 13.0 41.36 46.94 .2.24 9.18 11.97 4.43 16.40 36.22 17.7 63.92 9.17 3.82 12.99 Ig-285 21.06 8.97 30.03 36.22 17.7 63.92 14.88 5.82 20.70 Ib-285 36.22 17.7 53.92 17.89 7.33 25.22 6.94 2.24 9.18 Ic-285 28.36 13.0 41.36 9.17 3.82 12.99 4.89 1.44 6.33 20.3 8.57 28.87 24.60 10.9 35.50 8.27 2.79 11.06 Av. Hys. 17.72 Edd. 7.42 Total 26.14 Average losses per 511. cm. of tower 5.94 2.24 9.15 54.53 29.5 54.53 5.94 2.24 9.15 14.88 5.82 20.70 54.53 29.5 84.03 14.88 5.82 20.70 I‘a36o 28.36 13.0 41.36 45.44 23.5 68.94 21.06 8.97 30.03 Ib-350 49.72 26.26 75.97 21.06 8.97 30.03 11.97 4.43 16.40 15‘350 36.22 17.7 53.92 14.88 5.82 20.70 6.94 2.24 8.18 26.41 11.35 36.76 36.45 17.8 54.26 '11.91 4.41 16.32 Aw. Hys. 24.69 Edd. 11.19 Total 35.78 Average losses per 511. cm. of tower 14.88 6.82 20.70 75.01 43.9 118.91 9.17 3.82 12.99 28.36 13.0 41.36 86.40 52.4 138.80 23.76 10.04 33.79 I‘u445 54.53 29.5 84.03 75.01 43.9 118.91 23.75 10.04 33.79 Ib-445 86.40 52.4 138.80 36.22 17.7 53.92 14.88 6.82 20.70 10:445 28.36 13.04 41.36 21.06 8.97 30.03 6.94 2.24 9.18 39.73 19.85 59.58 56.25 30.65 86.90 14.99 5.87 20.86 Av. Eye. 36.99 Edd. 18.79 Total 55.78 Average losses per The following curves are 1055 distance curves. They show the distribution of loaaes at various points along the legs. The area under each curve was measured and when divided by the length it gave the same losses as obtained by computation when using the average flux. ,,__- .. Hysteresis Loss on Log A “1111 “!' O H . . ‘ : ‘H’II‘ ... n. 4 "O‘~:. i ." l-e. :_ 11" 0000 ~OO¢o v-o ,_,. o 944....A 4 o ’ ._ vn~4..... 4 "9 '9 9 o 0-19—6 1 - . .4 ‘4- . Ofooooao-A 000‘. o ‘140-v~ " “*-o—¢~—. . . rof:"’.‘ 9-4vo«o. 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B“ WAVE SHAPE m C - Wave: sHflPF m ‘ E.M.5 3- 16-28 Ln oecillogrem of the induced currents on the three legs. 75 Results from experiment (152)“ - ____§__e (230) 55.95 n (2.556303 - 2.361728) = 1.934498 - 1.591065 .194575 n - .343433 n a 1.765 (4491” - 120 (360) 86 n (2.543453 - 2.555303) 2.082785 - 1.934495 .087150 n .148287 n ’ 1.701 (15.9)“ - .___9__12 (230) 38.95 n (2.645453 ~ 2.361728) 2.082785 - 1.591065 .281725 n .491720 n - 1.745 g': U! (47 )n .___. (60) 49 n (2.575594 - 2.555303) 1.905455 - 1.590195 .120391 n - .218289 n . 1.813 5"“ __8_1_. L_Z§ (32 o) 42 n (2.575594 - 2.507555) - 1.905455 - 1.525249 .168838 n .285236 n a 1.689 Average n=(1.765 + 1.701 + 1.745 + 1.513 + 1.559J1/5 I 1.742 Reducing losses to same currents at different distances 1.742 (10g 350 - 10g 285) : 1.742 (2.556505 - 2.454845) 1.742 (.101458) .176740 10g 1 X 1.742 (10g 350 , 10g 200) - 1.742 (2.555303 - 2.301030)= 1.742 (.255273) .444685 log x I 1.742 (10g 360 - log 285) a 1.742 (2.555303 - 2.454545) .176740 log x I 1.742 (10g 350 - log 200) .444685 log 1 - I: log 86 - log 1 - 1.934498 - log 1 - 1.954498 - 10g 1 1.934498 - 10g x 1.757758 57.247 log 86 - 10g 2 - 1.984498 ~ log 1 - 1.934498 - log I 1.934495 - 10g x 1.489813 30.89 10g 49.22 - log I - 1.692142 - log x - 1.692142 - log x - 1.515402 = 32.76 log 49.22 - 10g 1 1.692142 - 10g 1 1.247457 17.67 76 1.742 (10g 440 - 10g 350) a 1.742 (2.543453 - 2.555303) 1.742 (.057150) .151515 log 2 I 1.742 (log 350 - log 285) .176740 a logx-a x: 1.742 (10g 350 - log 200) .444685 10g 1 x. 1.742 (10g 440 - 10g 350) : .151815 - 10g 1 x. The combination of those results are shown 77 log 2 - 10g 49.22 a log 1 - 1.692142 log 2 - 1.692142 - log 3 - 1.692142 1.843957 69.81 log 1 - log 25.14 10g 1 - 1.400365 1.577105 87.77 1.577105 - log 1 1.132420 13.56. log 2 - 1.577105 10g x - 1.577105 1.728920 53.57 in the following table. 78 Reducing to Pour Different Currents I Dis- Losses Dis- Losses Dis- Losses tance 104W tance 10-4w tance 10.4w 350 19.5 86 24 49.22 26.5 37.77 285 19.5 57.24 24 ' 32.76 26.5 25.14 200 19.5 30.89 24 17.67 26.5 15.56 440 19.5 120.55 24 59.51 ' 25.5 53.57 Four different currents were chosen in such a manner that the losses due to one of these currents are known at one distame while those due to the other are known for some other distance. By using the formula ( I 11.742 ‘ I ( Ia) L2 the losses for the various currents were found at different distances. 79 Fran the Table we get I 19.5; - 49,22 ( ( 24 86 x(1og 24 - log 19.5) = log 86 - 10g 49.22 x(1.380211 - 1.290035) 8 1.934498 - 1.692142 .090186 1 I 2.42356 x a 2.687 x (19.5) - 17,57 ( 24 ) 30.59 .090186 1 = 1.247237 - 1.489818 1 = 2.689 (19.5)1 . 32.75 ( 24 ) 57.24 .090186 x = 1.757758 - 1.515402 .090186 I s .242356 3 . 2.557 (19.5)x . 37.77 (25.5) 55 x[1.423246 - 1.290035] 8 1.934498 - 1.577105 .133211 x = .357393 I = 2.683 (19.5)1 . 25.14 (25.5) 57.24 .133211 I = 1.757758 - 1.400365 .133211 1 = .357393 x c 2.683 80 (19 5)! - 13 55 (25.5) 30.59 .133211 X = 1.489813 - 1.132420 .133211 I a .35739 x . 2.683 13.55 . ( 24 )x 17.57 (25.5) 1.247457 - 1.132420 = (1.423246 - 1.380211)x .115037 s .043035 I a 2.673 From the preceding results it is seen that x has a constant value. Therefore x = (2.687 + 2.689 + 2.687 + 2.683 + 2.683 + 2.673L1 6 x = 2.683 81 Deduction of laws From results it is seen that for a. constant distance (.IJJ“ - JA— (1) ( 12 ) L2 or more exactly (#11342 s__Ifl__ ‘2) 1 12 1 L2 and for a constant current (422.1” - .112. (a) 1 D2 1 L1 or more exactly 2.683 (.21.) ' .142. (4) ( D2 ) L1 Laws (2) and (4) give the relation between currents. distances, and losses Thus 22:?- 2—2-7 C. 5 4 - i 4 (6) showing that the same loss is obtained when current Il becomes 12 if D1 becomes D2. The value of D2 is foxmd from (5) or (6). 82 From the deduced laws let us find the losses due to a unit current at a unit distance. First find losses at distance 19.5" due to unit current. From (2) 1.742 L4LL_) =._Li_ ( 12 1 L3 ( 200) 30759 x 10"4 log 200 - 2.301030 1.742 10g 200 . 4.005394 (200)1'742 = 10145 1 = x 10145 30.59 10"4 x = 3.045 10'7 Then find losses at distance of 1 inch From (4) ( 1 ) 3.045 10‘7” 10g 19.5 - 1.290035 2.553 log 19.5 2 3.451153 2.683 (19.5) '3 2891.8 2891.8 - x 4 3.045 10 7 x - 8.805531 10.-4 ( 1 )1.742 g ( 350 ) 55 log 360 = 2.556305 1.742 log 360 - 4.453079 (350)1-743 ='25354 .__;L___-:._34_. 25354 55 x - 3.029 10'7 Then at 1 inch distance it is (19 5)2.553 . x _ ( 1 ) . 3.029 10'7 2891.8 = x 3.029 x 10‘ x - 5.759 10'4 ( 1 )1.742 ' x ( 255 ) 57.24510‘3 log 255 = 2.454545 1.742 log 285 3 4.276339 (255)1°743 = 15595 1 a x m—fi 18895 57.24 10 x a 3.029 Then at 1 inch distance it is ( 1 ) 3.029 10 x . 5.759 10"4 The average x is then 8.759 x 10-4 N] J, 83 Therefore, if it is desired to find the losses due to a current 1, D inches from the tower use the following formula LT . 5.759 10'4 (111-742 (D’sa As a check assume I t 440 D8 24" LT = 5.759 10"4 (449)1-742 (34) - log 44o - 2.543453 1.742 log 44o - 4.504595 (44o)1°742 . 40252 log 24 . 1.380211 2.553 log 24 - 3.703105 (20558:5 . 5047.5 LT . 5.759 10'4 49252 5047.5 a 5.759 x 7.97 x 10‘4 4 a 69.80 x 10- as checked with 69.81 x 10-4 ”TIT/117111)) ((M1)?!) (WES