A Determination of the Deviation of the Calibration Constant of a Force Voltmeter from its Theoretical Value for Isolated Spheres Due to the Presence of Other Bodies Citation Vaile, Robert Brainard, Jr. (1936) A Determination of the Deviation of the Calibration Constant of a Force Voltmeter from its Theoretical Value for Isolated Spheres Due to the Presence of Other Bodies. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8ajh-qt69. https://resolver.caltech.edu/CaltechTHESIS:11062024-212741052 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) 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 1936 Record Number: CaltechTHESIS:11062024-212741052 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11062024-212741052 DOI: 10.7907/8ajh-qt69 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16836 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 13 Nov 2024 18:17 Last Modified: 13 Nov 2024 18:26 Thesis Files PDF - Final Version See Usage Policy. 20MB Repository Staff Only: item control page

A DETERMINATION OF THE DEVIATION OF THE CALIBRATION CONSTANT OF A FORCE VOLTMETER FRClvI ITS THEORETICAL VALUE FOR ISOLATED SPHERES DUE TO THE PRESENCE OF OTHER BODIES Thesis by Robert B. Vaile Jr. In Partial Fulfillment of the Requirements For the Degree of Dootor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 1936 TABLE OF CONTENTS SUMMARY OF RESULTS INTRODUCTION Page l CERbNOLOGICAL OUTLINE 3 DESCRIPTION OF THE WORK 5 PART I Determination of the Effect of Shank Size by Capaoitanoe Measurement 5 Description of Apparatus 5 Description of the Tests 9 Results 13 PART II Determination of the Effect of Ground Planes by Measurem.ent of Foroe and Voltage 16 Description of Apparatus 16 Description or the Tests 19 Results 21 PART Ill Analytical Treatment of the Effect of Grounded Bodies 24 Force Between Two Isolated Spheres at Spa.oing of l Diameter 24 Effect of One Infinite Ground Plane at 5 Diameters 27 Effect of Four Ground Planes 31 Effect of Spherical Cage of 10 APPENDIX I APPENDIX II Diameters 37 Etteot of Spherical Cage of 20 Diameters 41 Results 43 Typical Data for Determination of the Effect of Shank Size by Capacitance Measurement 41 Calibration of Transformer by Sphere Gap Fla shover 47 APPENDIX III Typical Data for the Determination of the Effect of Ground Planes by Measurement of Forae and Voltage 51 SU~ ARY OF RESULTS The quantitative results of this investigation are: l) The effect of shanks, the diameter of which is 5% of the diameter of the spheres of a Force-Voltmeter, is to decrease the calibration factor k, in V • k 'IF, by 1.5% i 1% for spacings between 70% and 110% of the diameter of the spheres. 2) The effeot of a grounded, open-ended enclosing cubical cage, each face of which is 6 diameters from the spheres, is to increase k by 10% i 2% .for spacings betv."een 40% and 100% of the sphere diameter. The qualitative conclusions are: 1) The method of calibrating a Force-Voltmeter by me.king capacitance measurements appears feasible. 2) The presence of high potential bodies other than the sphere and nearby portions of its shank, is to decrease k by importe.nt and unexpected amounts (up to 15% at 100% spacing). LIST OF ILLUSTRATIONS FIGURE • l" 1 - Arrangement of Spheres., - Shanks in Place 2: Page 7, FIGURE 2 - Wiring Diagram 7 3 • Arrangement of Apparatus 7 FIGURE 4 - The Effect of Shank Size 14 FIGURE Model Voltmeter FIGURE 5 - FIGURE 6 - Details of Arrangement I FIGURE 7 - Model Voltmeter on Tower 17 17 17 FIGURE 8 - The Etfeot or Ground Planes 22: FIGURE 9 - Images For Four Ground Planes 34 FIGURE 10 • Working Graph of Capaoi tance Measurements 46 FIGURE 11 - Calibration of First Transfonn.er by Sphere Gap Flashover 50 FIGURE 12. - Working Graph of Force and Voltage Measurements 53 l INTRODUCTION. The force voltmeter experiments, made in the High Tension Laboratory as a part of the high voltage measurements program proposed bY, Professor (1) These Sorensen, consist of measurements with two one-meter spheres. spheres are so mounted that the one oonneoted to the unknown potential is stationary, while the other, which is at ground potential, is free to swing along the line between them. When a voltage is applied, the attractive foroe moves the grounded sphere toward the other one. If this attraction is now balanced by a mechanically applied force which returns the sphere to its original position, the attractive force oan be measured. To determine the value of the applied potential when a knowledge of the attractive force and of all physical dimensions is oomplete, it is necessary to know the form of the function relating these quantities. From three fundamental concepts: 1) that the foroe bet'lreen point oharges i s F • qlq2 r2 2) that the charge on eaoh electrode is q•C~ , (~ and 3) that the relative distribution of oharges over the surfaces or the electrodes is independent of the voltage; it is olear that the funotion relating force and voltage, when all physical dimensions are held constant, is of the form F • Kv2 or V = k '{F. latter equation k is a £unction of the physical dimensions. In the Primarily, it is a :function of the ratio of spacing (minimum distance between the spheres) to diameter, but it also depends in some measure on the position of other surfaces at ground potential and at the high potential. l. J.E. Hobson Ph.D. thesis 1935. "The · Sparkless Sphere Gap Voltm.eter 1', by Sorensen, Hobson, and RamoJ Electrical Engineering, June 1935. 2. It is recognized that the oharges on the two electrodes are not equal. The charge on the high potential sphere will be V tL~es C and that on the grounded sphere will be V times C where C and C are Maxwellian coefficients of capacity and induction. The important thing here is that charge is proportional to voltage, and that the factor of proportionality depends only on physioal dimensions. 11 1 ~, 11 1 ~ 2 Lord Kelvin has published tables giv~ng values of k as a. funotion spacing (3) of df t& for tv,o isolated spheres. ame r The values given are easily checked by the use of a few terms of the infinite series of image oharges < in the region spacing ! • where convergence is rapid. diameter 2 In &IJ¥ actual arrangement it is impossible to isolate the two spheres for there are, in the vicinity, other surfaces at ground poten• tial and at the high potential. Hence the use of Kelvin's values of k, with measured values of f'oroe will give only approximate values of . voltage. These other surfaces are primarily, 1) the shanks of the two spheres, and 2) the walls, floor, and ceiling of the laboratory. This olassifioation is made because analysis of a qualitative type indicates that the effect of both shanks is to decrease k while the effect of all (4) grounded bodies is to increase k •. The work to be described was undertaken to investigate the deviation of the constant k in V • k VF from its value for isolated spheres. when the effect of shanks and grounded bodies is included. Kelvin, "Papers on Eleotrioity and Magnetism."• p. 96. does not give "k~ itself, but rather 1 2 , {which he calls ttA1'). 3. k Kelvin It should be noted also that he uses the ratio of center-to-oenter distance to radius, instead of the ratio or spaoing to die.meter. 4.· The Effect or Shanks: An:s- 'Iiiorease inthe size of the high potential electrode (with V constant) oocasions more charge on it, and hence more force. Similarly• any increase in the size of the grounded electrode increases the oharge on it and also the foroe. The Effeot of Ground Planes: Any ground planeparallel to-the line of centers of the spheres mAy be replaced by image charges behind it. When these image charges are re-Smaged baok into the two spheres, it is found that the charge on the high potential sphere has been slightly increased while that on the grounded sphere has been deoreased by exaotly the same amount. Since the charge on the grounded sphere was originally the smaller, the produot of the charges after this addition and subtraction is always less tha.n the original product. That is (A + c)• (B - e) < AB it B <. A. If the ground plane is perpendicular to the line of oenters, the re-image oharges will not be exactly the same, but for all positions of practical importe.noe (say not nearer th.an 3 diameters from the nearest sphere), the difference in these re-image charges will not be sufficient to afteot the result already stated, namely that the force will be decreased. ~ CHRONOLOGICAL OUTLINE. The work reported in this thesis, as a part of the California Institute high voltage measurements program, was started in June, 1935 in the Institute high voltage laboratory and continued there until September of that year. From September until Apr il, 1936, the work herein reported was performed at Iowa State College, Ames, Iowa. After oheoking one of Kelvin's values for isolated spheres, a similar computation was made in which the effect of a ground plane parallel to the line joining the sphere centers was included. From this computation it appeared that the calculation of the simulte.neous effect of four or more ground planes would be vastly more complicated but not out of the question. Leaving this computation for a more auspi• cious time, experimental work on the same problem was carried on during the summer of 1935 in the High Tension Laboratory. The results indi• oated that it would be impossible to determine experimentally the effeot of ground planes without some knowledge of' the effect of shanks, and vice-versa. As an incidental result, some doubt was raised regarding the applicability of Peek's flashover values for 12.5 om. spheres to the test.s ma.de in 1927 and repeated in 1936. (See page 47.) During the fall of 1935 the calculation of the simultaneous effect of four ground planes was resumed. A double infinite series was devel• oped, which was proven convergent but which was found hopeless from the standpoint of oompute.tion. for 1% accuracy. Approximately 10,000 terms would be required (See page 31 for details of the method.) Attention was then directed toward an experimental determination of the effect of shank size. In considering methods, several possi• bilities presented themselves. First there was the direct method of measuring force and voltage with different sized shanks. A setup was made following this method, but unusual difficulties in measuring both force and voltage were encountered and it was abandoned. Another 4 possibil ity was the use of an eleotrolyt~o setup. Conduoti on currents in suoh an arrangement are equivalent to capacitance currents in the electrostatic oase. The electrolytic experiment has an important ad• vantage in that perfect insulators are available. It has also, however. the serious disadvantage that the specific oonduotivity of solutions 5 varies extraordinarily with temperature. ( ) For this reason, the electrolytio experiment was not attempted. measurement of oapaoitanoe. A third possibility lay in the This may be shown as follows: In any arrangement, the energy in the electrostatic field · is 1 ...a Ef•fCv ·• It, now, the capacitance is increased by dC (by reducing the spacing) while the voltage is mpt constant. the increase in the energy of the electrostatic field will be l . dEr •·· 2 ·v2dc. How.ever, the source must have supplied energy amounting to dEs • Vdq • v2°dc. Sinoe only half the energy supplied appears in the eleotroste.tio field, the remaining half must appear as work done by the system on the external world in ohanging the capacitance. 1 l 2 Thus dEm - dES - dEr .. v2dc - · 2 v2dc • a: V dC. If this change or capacite.noe·oocurred by the change of a linear dimension, x, then Fx • - dF.m 1 ~• - a _l__ r cbtdC Or• in V • k VF, k • v1 ac • 2 dx Hence, if, tor a given shank size, the curve of capacitance against spacing can be detennined experimentally, the value of the constant relating voltage to the equare root of force is dete~ned. usually good equipnent for such cape.oitanoe measurements was available and a complete set of tests was run. 5. Approximately- 10% per 0.1°c. ~ »ESCRIPTION OF 1'HE WORK ------The following detailed desoription of the work will be divided into three parts: fi.rst, the determination of the effeot of shanks by oap,.o- i'tanoe measurements, BGoond, the deternd:nation ,of' the ef'teot of' ground planes by measurement ot toroe and voltage, and thitd, the analysis of the etteot or ground planes by the method of images. PART I Etfect ot ~--Shank -The Determination of the --- . , - - - - -- ;.,...., , . - -- . ·-' - - It is telt that the importanoe or the method tar transcends the results so tar attained by its use. By means of it, the final c,alibration of a~ force-voltmeter may be obtained, with no regard tor si~e, posi• tion, or uniformity of' ground surfaces or shanke (except that the t o ~ . . tion of' corona must be avoided). Too much emphasis cannot be laid on the point that a calibration, determined by measurements on the actual meter in its permanent place, is n.stl;r superior to one whioh is caloulated from figures for an "ideal ff arrangement modified by correction factor• determined by a_n,l_ method. Description of the Apparatus Spherical• copper-plated, pressure noata, 3• in diameter, were 1• 1• used with shanks ot 4 , a: , and l ". The arrangement is shown in Figure • 1• 1, page ?e The floats were tapped tor the 4 shanks:• end these shanks, both so• long, were used to support the spheres during all tests. When 1• measurements ot the larger shanks were to be made• sleeves of i (or 1•) 1" permanent shanka outside diameter and 12• long, were slipped over the i and oentered by means of aoourately turned washers. Between the t.o 4 • glaz.- od porcelain insulators supporting one shank,• wooden engineer's-aoale ti was fastened so that one edge closely par~lleled the shank. On this shank a vernier scale was scribed so that the position or the sphere oould be determined to o.oos inohes. The circuit used in measuring the oapacitanoe is shown in Figure 2, page 'T. The general scheme was to place a vernier oond~nser in parallel with the 1pheres, of which the oapaoi tanoe change was to be measured• in. one arm of a oapaoitanoe bridge, and to plaoe a fixed oapaoitanoe in the other arm. J:ny change in the -cape.ci tanoe ot the spheres must then be 1 equalled by an opposite change in the oapaoitance ot the vernier oondeuer ' before the bridge is restored to balanoe. The fixed capacitance was a General Radio No. 222 precision variable ) condenser. Its setting was kept unchanged tor the duration of the tests. The vernier condenser was made by the Iowa State College Instrument Shop. It consisted or a stationary vertioal outer oylind•r and a oon- oentrio inner o;ylinder mounted on a micrometer lead screw having a pitch ot .s mm. per revolution. The oiroumterenoe ot the head was. divided into 25 parts and the vertical travel was 50 mm. ao that the total range ot oa.paoitanoe wa.s divided into 2500 divisions. The total range was found to be about 220 Mioro-Mioro-tarade so that each division wa.s approzi• (6) :mately 0.1 m.m.t.. •• Further. each divi~ion on the condenser covered a apace of about 3 mm. ao that tenths could be easily estimated. Bence ,alues of oapaoi~noe were recorded down to 0.01 m.m.t •• The taot that the ourves obtained a.re extremely amooth ie good evidence that the figures are sigmtioe.nt to this precision. The oa.paoitance bridge• a General Radio No. 216 precision type was completely shielded and had shielded transformers in both the osoillator oirouit and the output circuit. The fixed relJiatance arms had 5.000 . • t, ' ' . -- ohms in each and the variable resistance (deoa.de type., from 10000 ohms s. The abbreviation "ia.m.t." will be used tor mioro-micro-f'ara.d hereafter. 1 FIGURE l 1• Arrangement ot Spheres, 2 Shanks in Place STEAM To PIPE " O SCILLATOR, FIGURE a Wiring Diagram FIGURES .Arrangement ot Apparatue 8 to 1 ohm) could be switched into either oondenser arm. The oscillator was a Western Eleotrio type SA, audio-frequenoy oscillator. It was about 40 feet tr0111 the bridge. in a separate ro0t11 and on the floor above. A trequenoy of 1000 cycles was used thruout the tests. Power from the oscillator was led to the bridge over a twisted pair. It was round experimentally tha.t a sharper balance could be ob- tained it a shielded transformer, with ungrounded oaee, was plaoed in the oscillator leads, close to the bridge. "Amertran" Mixing Transformer. The transformer used was &n A similar one was also tried in the out- put circuit but since no improvement was observed, it was removed. The amplifier was a Leeds and Northrup No. 210301 two stage audio- frequency amplifier which could be broadly tuned tor 500, 1000 or 2000 oycles. The tubes used were one 201.l and one 212, which were so ex• tremely miorophonio that the whole amplitier was plaoed on sponge rubber pads on a separate table. The amplifier was completely shielded, and this shield, together with the shield in the bridge, was grounded to the at-.m pipe. Early in the work it was found necessary to bridge a 0.02 m.t. condenser across the input to the amplifier in order to eliminate in• terterenoe from the broadcast transmitter located in the same building. Thie condenser,. ha'ring an impedance of 8000 ohm.a at 1000 oyoles, un- doubtedly reduoed the sensitivity of the bridge but the amplifier more than made up the ditterenoe. As a matter of' tact,. it may well have been that the condenser reduoed the sensitivity greatly when the bridge waa I tar out ot balance and had progressivoly less etteot as the bridge was brought to balance. control. It would thus act somewhat as an autanatic TOlume In l.'ZJ¥ oaae, a muoh o1oaer and more near~ reproduoable ba.lanoe could be obtained with the use of the amplifier and condenser than with.out them. The phones were Western Electric No. 609-W. A photograph showing all the equipnent with the exception or the 9 amplifier. which is just out of the picture in the right foreground• is included as Figure :s, page 7. Description of the 'teats The calibration of the vernier condenser was considered of paramount importanoe and hence it constituted the first task. For this work the General Radio No. 222 variable condenser was placed. in parallel with the ~rm.er condenser in one arm of the bridge, and a third condenser. the setting or which was kept constant thruout the oalibration, was plaoed in the other arm. After a balance had been obtained, the No. 222 con- denser setting was changed and the bridge was rebalanced by adjusting the vernier condenser. Thia process was repeated for about 50 points during which the complete range of the vernier was oovered several times. It was found that the w ,: rnier cond,nser deviated measurably- from a straight line tor the first 20% ot its range, but tor the middle 60% out of a total of 43 points taken, 2 deviated from the best straight line by l.5m.m.t. and all the others were within 0.1 .m.m.r.. . The range cov- ered by this straight line portion was from 183.3 m.m.t. at setting 2000 • to 326.6 m.m.t. at setting 500. Henoe l di'Vition on the vernier ia S25.6 • 185.~ • o l'\ftA95 2000 • 500 ♦ V;;,,a ffl ~ WLf -- U From the bedmd.ng ot the calibration work i t waa apparent that the General Radio No. 222 oondenaer had a notioeable blaoklaah. Ba.cJD. lash is,. of course, undesirable in aey precision equipnent but it ia particularly unf'ortunate and annoying when a null method is employe.d. To eliminate the etteot of this backlash. at each point the bridge waa unbalanced by moving the No. 222 oondenser in a specified dirsotion, and the rebalance was effected 'With the vernier condenser, which had no detectable backlash. To inoree.ae the accuracy further, the No. 222 condenser was used only- within the range where its calibration was an exact straight line. Temperatures were recorded, and varied samewhat 10 tram day to day. No correlation between temperature and calibration was found. The procedure in conducting the tests was as follcrwa. diagram refer to figure 2:. page 7.) (For wiring The position of the le:rt-han.d 1) sphere was adjusted sQ that the desired range ot spacing could be at' tained by motion of the other sphere alone. a) The right-hand sphere as then a.djus~d roughly to any convenient spacing (say 0.1") and its position., as , read on the engineer's scale and the vernier scribed on the shank. was recorded. S) The bridge was balanced with the vernier con• densei- and its reading i-ecorded. 4) The right-hand sphere was moved to increase the spacing approximately one half inch, its position recorded and the bridge rebalanced with the vernier condenser. This process was repeated until the desired range of spacing had been covered, from. 3" to 4.5•, (that is• from 100% to 150% ot the sphere diameter). The right- hand sphere was then returned to an initial spacing of approximately 0.2" and readings were taken at half' inoh intervals as before. The sphere was then returned to o.3"'• o.4tt, and o.s" initial spacing and the process repeated. It should be noted that while the ad;j!lstment of the sphere at eaoh point was only approximate, its £<!sit~~~ was imown and recorded to o.oos". At the end of a complete series, the eJnplifier was turned oft and the right-hand sphere was carefully moved toward the left. When 1t touched the other sphere, the 'bridge we.s violently unbe.lanced and the 1000 oycle note could be heard. The position of the right-hand aphere ,vas recorded, and this reading "was subtracted from a.11 points to get the epaoillg• The results were plotted as a curve of wrnier setting against sphere position on which the points nre approximately 0.1" ape.rt, though the range had been oowred, not once, but five times. This process was followed to eliminate any changes in condenser calibration or. in tact. ~ error which might be changing continuoualy thruout the run. When suoh changes ooourred• as they did in Jll.8cy of the early tests. they were 11 immediately detected by the appearanoe o~ "waves " having a wavelength of 5 points, · in the final ourve. It was round that keeping the room temperature oonstant within, say, 2 degree c. for 6 hours preceding each test and during it, eliminated the difficulty. This indioa.tes that there was a variation of some capaoitanoe with temperature, notwithstanding the contrary oonolusion reached during the calibration, in regard to the vernier condenser. Rough calculations immediately eliminate the possibility of the expansion of the shanks changing the spacing of the spheres in suffioient amount, and while both the vernier condenser and the No. 222 condenser in the other arm are open to question, it is considered more probable that the vernier condenser was at fault. The effect was so small as to be negligible as far as calibration was oonoerned while still of importance in making the position of the capacitancespacing curve doubtful. The data of a typioe.l run is included in Appendix I. dC To fi~d di from. curves of vernier condenser setting versus sphere position. it was necessary to oonstruot a ourve with ordinates the slopes of the original curve, and then to apply the calibration factor of the vernier oondenser. These slopes were found by means of a plane meta.llio mirror about 10" long, whioh was plaoed across the curve so that its reflecting face was nonnal to the plane of the paper. The horizontal angle between the mirror and the curve was then adjusted by rotating the mirror, until the curve on the paper and its image in the mirror had no discontinuity of slope at the mirror edge. If the observer now views the curve and its reflection at an angle of 5 or 10 degrees to the paper. the position of the mirror oan be determined with ooasiderable precision, and a line can be soribed using it as a straightedge. From this line, which is normal to the ourve, the slope or the tangent can be calculated. The values of the tangents found by this process were plotted on the same sheet with the original curve of vernier condenser setting versus 12 sphere position. The final sets of data. taken after several weeks ot trials, oon- I" l" aisted of two oomplete runs for each size or shank• 4 • _ • and l". 2 Ea.ch rUD consisted of points at 0.1" intervals over a 4.5" range ot spe.oing (150% ot sphere diameter). The values of slope from the two curves for one shank size were plotted on the same sheet, and a smooth ourve was drawn thru them. From this curve values of kwere computed as follows: In V • k VF. k should preferably be of suoh dimensions that with F in grams. V will be in k:ilovol ts. E • When, on page 4, the equation ~ C~ was written. the uni ts implied were E • energy in ergs C: oapaoitance in stat-farads(• centimeters) V • voltage in stat-volts. '\<'Vhen the derivative with respect to distanoe is taken• dE 1 _--de F • - dx • - 2 V"'h• where F is in ~nes if x is in centimeters. Dimensionally. this equation is• then• n..........es • 1 ( a ta...-vo - lts )• • stat-farads .,.,. cm. 2 But it is desired to have: Fin grams Vin kilovolts : in micro-micro-farads per cm. If the symbols F• v, c. and x are re-defined in these units. the dimensional equation above must still be satisfied and there results F • 980.S • i (~•-• 10(.2998) f From which v _ 980.6 X 2 X (.2998)8: a 14.00 k•'/F - 10(2998)1- dC di I~ V"crx -. Thia is the constant of general applioability but one further step will 13 be required to find kin terms of the ordinates ot the last curv• plotted from experimental data. The ordinate of this curve (oall it "N") is in vernier divisions per inch. It was shown on page 9 that l division • 0.09495 m.m.:r., and l inoh • 2.54 om.J hence dC • N x (0.09495) cbt 2.54 .And k • ~. 14.0 • • 72.4 ,r,; (0.09495)i .,,-I ... 2.54 Values of k tound in this manner tor the 3 shank sizes are plotted S~oing against Diameter in curves c. D, and E of Figure 4. Results or Capacitance Measurements to Determine --~ Ef'feot of S ~ Siz~.• The results of the tests described are shown graphically in Figure 4, page 14• on which values of k, in V • k (i, are plotted againat the ratio or Spaoing to Diameter. Curves C, D, and E represent the experi• l", l"' , and l" respectively. mental results tor shanks of i 2 (Since the spheres were 3" diameter these shank sizes correspond to s.S3%, 16.67% and 33.33% or the sphere diameter.) Curve A 11 the theoretical curve for isola.ted spheres (from Lord Kelvin's data, footnote 2, page 2). Curve B is the extrapolation of curves c, D, and E to ,z ero shank size. The comparison or ourves C, D. and E with curve Bis the essential result. From them it is seen that a shank size of 5% or the sphere I diameter (which is 2" for 1 meter spheres, results in a reduction or k ot approximately 2%. for spacings from 70% to 115% of the sphere diameter. If there had been no equipotential surfaces other than the spheres and shanks in the f'ield, curve B should have ooinoided with curve A. That they do not coincide results from the proximity of the -vernier condenser to the movable sphere. The case of this condenser was oon• nected to the stationary (left-hand) sphere. Hence when the right-hand sphere was moved, the resultant oapaeitance change had two parts, 1) the change in capacitance between the spheres, a.nd 2) the change in oapao• itano• between the right-hand sphere and the vernier condenser case. 14 " FIGURE 4 :,;,: _: . . :.. • · _ ' ·.. .. • .•· ,: :. _:.·_ • · · ·l• : .•· ·. - t ' · · ·· - - Cd I :.: . :• 1-- - i,· •.~·;· ·' · ·, \·. . l.•. - - I •1· -~ - -~--1 --; .._ ' :. r\ '.'~ 1• I·.::. ::_:·T: -.·.• ·,:_ ', . "· I , · , - t : - ·-·· ~ -- ,·, • 1 '··· , - 1 ' i..: ~. • : • - · :·-• - • • - : ;; .\ , •·:-r4i ........ - ·· .. -t:-- '--- - - - - -t .~ +·~.·.•, .·. ! •• ,.. . I, ' . i.: · :,., : I I 1··· --\ .' iL· 1 I •\ 1 , ·· ·--! - ··-; • • - • • • •1 ; /: . i" • • , . . ! ' I- ••• 1.j. I I. , - f· l I • · ! ,. I • , • I I r '. - - - 1 .. ' I • ~. , • I , •; 1 • .! .. ,. ' '.J~i ~T~J:i:+~[:-FF~Fl -~.:~:.;_;t' ~r·t.· Q ' • : I •• ; C., • lj _I ,;-~~01 ·~··i - J-, · :··~i·,. .• ·1 1 . I T I (·;t ~:"'.,: -,- r--~. ~ -~ -·-·r•1- •• 1 l.t ::~- ',.Jrl f J;--:+\ · :-i:·· ! )d -~ -- ·:~ • :: i ~' :1v f:-·+ ·+ --.: ·1·~~·-· :~--:~-T-_-i-1JxE;l~--:r~r~·77f.~'.T', ·:r-.-.:;R·; 'i. +-·_·t:-:'t.r. 7 .. ' .. 1..-·;1--• Q ·11- -c--' ~J.-~.:( 1 -~ -~- _: ~+-,:: ·-·~~ :}.--=-:..;.~-·-: --:-~--1.,.:i-·•+- - - 1 ~~- f~--~-·· ,!' • 1 'i ' I. j . I ; l " : ~1:- I l ::~·1 : ;--: ,T. -r, ? .i r-l ·:· -:- r·.,t-~-t~.---:·--~r r· ~i~-- - d ~·-1-:- r : : ~:r.~L-·--:·~·fr-·/t ;- ;..J-!.""~~-x: -, · ·--i,-·I. ! --~ - - t~?~ • ., ••, •• ' • • • ' ' I l I • • • • I- . I ' • ' ~ ~ • ·.-··t·· , ·-.··-:-.- ~I:-:-~.-:· l·:T-=:,, ·_·,--:-: . :_r ..-: •• i•---r··-: 1::: 1 If\; • 1 -I · ! : · ' ·,· I - ·: :. I .. 1 .. ' •• 1 ,· •• V: ~ - - · ' ·· ···--'rf-.:..+--· ~1.1 ·•__ --' · · 1- ··i--- •-·•-,.:. __ .L_.. _ f , - -- :: • ..._; """ ·,, ·i, · ·- · , _ I,: i• 1' ,·,. ·f •...~· I .·,I. ·1 •· 1 · I 1• j • I 1·· ' .---1 -,- -• ··, -;- --··· ·-: . -·--·-. ----- , ·I-- , .. ·-- ··--l··· ' - 1 l •; 1 1 >1 , : ,.: · : . . ; 1 : ·- 1• ; ! _, _, __ , _ _ ,...L- -~ - · •· -- 1 (· '. ) ! 1 l • '. ; .~ • ~: ~ 1 ' :L, [ f '. :••.•.f ··,. J·•· •t·-.i .}j •. ji;t_]'It. l :.• • ~ :..i : : , , ~, _ ; • - +'·-J.c -~·--1 _.'....'...+'... . ..'. .. -,. - ij ••--~-..;· i-.:., .. : - j • "j 1 d ;. ' - 1 I j ,ll: ~ . . . ttJ•·•t: ;,;,.;f~lL-j.ij t: ·----1- ...!. •• - .... -·~----e:, +··-L-i•-·:-l-1~-L ___ ,_ _ .cl., .. ,...... !: · 1 :· ;·:~·t_;_!·:::.: 1-;::··[:::.; ::~f .: :1 :!\·! l: ;:f!.1: :: _, :·~~~ ,>- . !_::: i , •,··r • ,,., ., . ··.·.1 • I -r 1•• L . -. •• ·•. -· . ~-- ··•··-.~ r~1;· >-q-: l ·~·- i.- ·,t- - ·j ·::· -! fJ.~~r·r. t:~·~··\-r-~-·-J_-· •-: 1-- ··J-.,., ••• c.. , ~L _ j ---t --'. -:-, ~ l\ . ·.- '.·: ij 1 ~-.(~.--- -~ l'; ·;- ·+·· · 1 ' ·-·· · ().> - { - \ j__ - !-'.=:·1 .-~ _: jL TT::·;, • · l •· I.-· +- ·-· -·T· ---.:-· . l·; __ ,'.~ l-- ·I:· '. I ., L )••• .~t~-, :,: ••• :r<'. t J ~ ~~~~~\~. t·~f ;: :·;--- ! ; ~ ·-: .-. :- I :.;.j .J IQ r:---· ···~-- •··•· •-···+···· ··1··•-- :--J •• _________ • • . : :._:.-:.:½!'•'.: ;·•·_:·'.·.· .~,;._· .:·._~:.1._:_•. _. li,:·•··:·, 1,2·..:...;l,.:..•. -.·_·_~:r·~.•·_Jll,_:-'.• _ti.:J_.~_:~~:::·'.-: _~_:_;(_- ;:,·,· .-_·~-,;!, ~ -~ . • '' • • I '' •• ' • • ' ! .. i ' I • ' I • L • l ' : -,~ --· vj---·.:.., .:Lc::....'..-~···- - +--· --1---~~ . . ·k_l ___ L -- ,;...:..:. _;.........r--'.- Y ~~ • • ~~ ' I' :ftr~ ·}~Jfi~'~j I· J(f {t.*TFi-f r 7 --.. ,7. t ·,·--:: ~ ·-u ~- -- . ....1- '7f-;..,._;--~--,·· ..:. ~ : '.·_ 1 I - \ft~j~~t:t~_t_j~_:f'..~T}~~ - :~I~JL{~l l • . ~ ,. ' !. I .< : "I.'' .... - ··t-·:- • -~----· -- . ·,'' ---- .....--- ---'' - ~-:~r _1 ; : _- : . ·f , I .'' 1 ! ~ -1: ~ ; r • 1 I -rj-~_;: ~ --: ' '... ~---1--· - ~~ j ·- r I .' l •• - · J.-:f::./-'.c:-'. I·--- ·'.·t_:::-i·-:-L--- · f· - -:- - ! I•. • r,:~-., -.~: ,., l_ , I . ••• ' '1 - • "' - ·1· ···. ': ···1; . . •. ,' : .. ,. -' -··· • 1··.· l.-~ i- ·····, i ;···- .1 I 1. i -- ! ! .• :• •: • .,. r , ! I •; • • •1 • · ·I· ' ' , . . : ..·J,. ·<.· · •, ·-'I. :: ·. ' • ·1. •,_.=>1~:.: .~. ---L -.L·-J •. ' ; --·,, --··· •:-.,,· !. ·,··).... ·.,: 1·...:.:!., . .. ,. . ' · •-'·· I · ·~-·---· ... ' -•-,· ... ·-1······ ·, .•. .. . i .... ... . _1 :· _ -~ __ -!_ _, •:; ~--!---·- -__ _ L :.~l------t-= I ·_ i: :- : ~: .::_l., ·t ,: •• • .; : ·:·i:• · i . .. ;:; : · ~ : .. . • •• ,·!;. ··:-· , . · _iI -·-: :" · 1: 1:1·+ .. . · · --···. ,,: ..;.... .l. - ~-! .:;.:_ " I - -·- ..,i .•··.· ·. ·. :. ·- · r- '· •l • '~=- i. • ·t . ··· · : :::~- .. f • : ·. '· • - j: _.-1 :1 T • -i I I I • -f 1 , ·- iJ .. ,: <L~T~: !.~ l .. J · !-' l .: j __ i: - i. :L. ! . ' l •. . . . : ' ! :. r, _ -·-1----1. ~- -T:~---r -.----.:, -~-..--T-_ • •• ... J • --~-~---! _i •- •:· ' :-• : )1· : ·- : ··, l • • •. -- ·- i I :_fj_\L_j __:~ I 7 -:.. 1- • _ L · :- • • • J • ·•i ..: ' ·: .i i' ~- ·: :_:Il ·.: ..:~;.·.::: __ : ~ 1! i : <;_::_--i.•.· _i. ·T < -_I :'.h~ · . L. ,: ;;~~: j ~ ...... 15 Since the right-hand sphere corresponds to the grounded one in an actual volimeter, oonneoting the vernier condenser case to the other sphere oorresponds to the addition of a large high-voltage body in the actual meter. Its effect, in this instance, well illustrates that bringing in any body at high voltage, other than the one sphere and its shank, changes the calibration of the voltmeter. The tact that the curves cross at about 55% spacing has not been explained. Nevertheless it is not believed to indicate solely lack of precision because the two curves run tor each shank are in agreement to s01J1ewhat smaller values. reproduoa ble to within lf'- It is estimated that these curves are 16 PART II THE DETERMINATION OF THE EFFECT OF GROUlID PLANES BY MEASUREMENT _2! FORCE ~ VOLTAGE. Desoription ~~ Apparatus The use of a model arrangement rather than the use of the 1 meter spheres was decided upon primarily beoause of the comparative ease with whioh different arrangements of ground planes could be set up. A pair of 12.5 cm. copper spheres which had been in use as a sphere gap were chosen. These spheres were set up inside a wooden cage 8' high by 1 OI long by 6' wide as ah own in Figure 5 • page 17. The high voltage one was mounted so that the spaoing oould be w.ried• but in adcli tion• so that when it was clamped in place its position was fixed. The 5"' 8 threaded and calibrated shank about 14.. long. 'W'h:1ch was attached to it in the sphere gap, was retained. The metal shank or the other sphere was replaced by one ot wood about 5' long. This sphere and shank were ,, mechanically suspended by a double nvn ,s~spenaion exactly similar to ' the large voltmeter, and the sphere was grounded by: a No. 20 wire running along the wood shank and connecting to a coiled pigtail at the rear "v" ~upport. Close to the rear end of the wood shank a brass ring was mounted, across the vertical diameter of which a 0.6 mil tungsten wire was stretched. This wire was observed with the oa't;hetometer to detect motion of the sphere. ing observations. A fluid damper proved to be ot great aid in speed- 'l'he vane was e.ttaohed to the shank, and the container. a cocoa oan, rested on the frame. The details will be made clear by a glance at Figure 6. In measuring the force it was considered better. tram the stand• point or aocuraoy and speed• to read the sphere deflection in the oe.thetometer and to calculate the force tram these deflections by use of a calibration curve. For this calibration, and also to prevent too l? FIGURE 6 Model Voltmeter FIGURE 6 Details or Arrangement FIGURE 7 Model Voltmeter on Tonr 18 large defleotions, a means 0£ applying a horizontal force was r equired. The method used was to suspend a weight from two equal threads, of which one was attaohed to the wooden shank close to the front "V" support, and the other to the stationary frame at the rear. Care was taken to plaoe the two points of support on the same level and to make the distance between them approximately equal to the length of each thread. From measurements or the three distances, the horizontal foroe due to aey weight oould be calculated. With this arrangement, horizontal forces from 1 to . 5 grams were applied to the grounded sphere and the deflections in the cathetometer were observed. The calibration ourve so obtained was aoourately linear. Though the weights were within only about 3% of their marked values, each one ,ms carefully weighed on a sensitive balanoe to the nearest milligram, and this value -was used in oalculating the horizontal force. The maximum actual defleotion of the sphere, if the cross wire was to remain in the oathetom.eter field of view, was 2 mm.• The souroe of voltage was Unit No. 1 of the One Million Volt Casoade transformers. 250 KV. This unit has a maximum seoondary voltage of It contains a tertiary (voltmeter) winding having a turn ratio with respect to the secondary of 1 : 833J ooil represented 250 KV on the seoondary. hence 300 volts on the tertiary Since these transformers were designed to have very low reactance (1.24% impedance by test), a series resistance to limit flashover current was an absolute necessity. 3 ,tt The resistance used was five feet of 4 garden hose filled with tap water (about a,ooo ohms per foot) which can be seen at the rear of the oage in Figures. Considerable care was used to be sure that flashover did not ooour• particularly after the first time, when it was discovered that the water column acted more like an expulsion cutout than like a protective resistance. The eleotrioal oage consisted of enameled iron fly screen in the !9 form of an open-ended cube. Each face was 29.6" (or 6 diameters ) from the centers of the two spheres. The end into which the high voltage lee.d entered was omitted. Description 2£~ Tests. At the outset of the tests it was recognized that if there was any uncertainty regarding the tre.nsfonner•s secondary voltage in terms of its tertiary ooil voltage, the results of the tests could be only com• parative; that is, only the percent change in k (in V = k '{F) when the grounded cage was brought up, oould be determined. On the other hand, if the true voltage applied to the spheres could be determined with satis• factory accuracy, the numerical value of k under each condition could be found. The unusually low reactance of the transformer should indicate the maintenance of -the nominal ratio between tertiary and secondary over the full load range within a fraction of l percent, and since the transformer load during these tests (simply charging current) was less than 100 m.a., compared to the rated current of 1 ampere, the ratio should have been maintained within even oloser limits. In addition, since the leakage reaotance is essentially independent of the flux density and since the charging current is directly proportional to the voltage (up to the beginning of corona ) , the IX drop should be exactly proportional to the voltage. Renae if the calibration curve of secondary KV against tertiary coil volts. deviates from the turn ratio at all, it should re:nis.i n a straight line with merely a slightly different slope. Since the charging ,current leads the voltage, the IX drop adds to the induced voltage and the final conclusion is, then, that if the ratio of secondary KV to tertiary volts, which is constant in any oase, differs from the turn ratio at all, then it must be higher by not more than 1%. The conclusion from theoretical considerations, that for any maintained arre.ngement the force between two bodies is exactly proportional 20 to the square of the voltage. seems to require uo qualifying limitations to its practical application. exoept. perhaps. the statement that the formation of corona is eleotrostatically equivalent to an increase in ' size of the oonduotor. Thus. with no knowledge of the numerical value of k. it was possible to use the model spheres to check the linearity of the ratio of secondary KV to tertiary volts. st The first tests indicated exact linearity between F and v , up to ,. . (7) v• 100 (V • 83.~ KV ) approximately. At about this value a slight but definite change in slope occurred corresponding to a reduction in k of 3 or 4 f~ After this break was observed it was noted that audible and visible corona occurred on the high voltage lead at v • 58.5 volts (V • 49 KV). and on the shank at v • 12-:3.5 volts (V • 103 KV). All tests run after this discovery were carried only up to v • 100 volts. The procedure on a typical run was as follows. With the cathe- tometer adjusted so that the wire was near the center cf the field and i ta reading taken, the sphere to be com1ected to the high potential was moved over until it touched the suspended one. The calibrated shank was read and the sphere was mowd away to approximately the desired spacing• when the shank we.s read again. After rechecldng the zero oathetometer reading. the voltage was applied, the oathetometer again read. and the tertiary ooil voltmeter read. The voltage was increased and readings ot ca.thetometer and voltmeter recorded again.. When the wire- approached the limit of the cathetometer field• a 5 gram weight was hung tram the double • thread support described on page 18• and the oathetometer read again before any further change in voltage was made. The voltage was then increased in steps until the wire had again moved to the edge of the cath.etaneter field when the 5 gram weight was replaced by one or 10 0 7. The lower case "v" wili be used for tertiary' coil voltage'; in oontradistinotion GO the upper ease "V" which will continue to be used tor kilovolts applied to the meter (that is, secondary KV). 2I grams. For short spacings it was sometimes necessary to make a third traverse across the oathetometer field with a total weight of 15 gre.ms. (The ratio of horizontal foroe applied to the sphere, to weight hung from the thread• was o.3 to 1. henoe the horizontal toroe due to a 5 gram weight was 1.5 grams.) Since no break in the curve of v11 against F was noted when the 5 grrun.s weight was added (thus changing the spe.oing slightly}. it was assumed that no correction for this change in spe.oing need be made. When the curve of T2with respect to F was plotted, the slope, with the assumption that the turn ratio between v and V was valid• gave a single value of k for the spacing used, and tor that particular arrangement or ground planes. The spacing was then changed, while keeping the arrangement otherwise constant, and a new value of k determined. Eleven runs were made, tram which eleven values of k at different spacings could be determined, for eaoh arrangement of ground planes. arrangements were investigated: Three general 1) no cage, the only close ground plane being the floor which was distant 7e85 diameters, 2) grounded cubical cage (except tor one end). each side being distant 6 diameters. 3) no oe.ge. apparatus being on a 'WOOden tower of suoh height that the floor was distant 30 die.meters. On thia la.st setup. it was noted only after the data had been taken that one of the l meter spheres, which n.s about 15 diameters to one side and about 6 diameters behind the model 'VOltmeter, had been connected to the high voltage during this test only. Results or the Determination of the Effect of Ground Planes -- ~ Measurements - ~ ---- ---- --- ·- of Force an<!_ ~.@:!'• The results are shown graphically in Figure 8 on page 22. On the eypothesis that the use of the turn ratio of 833: l is a valid means of determining secondary voltage from tertiary coil voltmeter readings. numerical values of k are significant. as is also the comparison of the experimental curves G. H, and I with the theoretical curve A• tor 22 FIGURE 8 13: isolated spheres. Evidenoe that the use of the turn ratio does not introduce very serious error is that a sphere gap oalibre.tion curve run by w. A. Lewis shows a deviation from the turn ratio or 1.s% at 70 KV. A simila.r calibration by sphere ga.p (described in Appendix 11) whioh was run at the conclusion of the force and voltage measurements shows a deviation of 3.5% at 70 KV. When it ia reoalled that eaoh "Value ot k is determined by the average slope ot a Voltage-Force curve on which nearly all the points lie below 70 KV, it is seen that these deviations would oause errors of muoh smaller amounts. The most interesting of the curves of Figure 8 is I, £or the tests made on the tower. During these tests the nearest ground plane, the floor of' the laboratory, was at a distance of 30 diameters. The all- important question is, why are the values of' k so far below curve A for isolated spheres (that is, why is the fo~ce for a given voltage so much higher than would be expected)? The explanation offered is that the effective high voltage electrode includes much more than the one sphere. It appears that the effect of the shank, the lead, the hose resistance. and the Orie-gallon can at its upper end• as well as the 1 meter sphere mentioned earlier, was more important than had been supposed. This conclusion is fully born out by the similar results of the capacitance measurements. discussed on page 13. Curve H• in which the disturbing eft'eot of the 1 meter sphere ia absent. and in which the ground ple.ne (floor} at 7.85 diameters tends to increase k• is still below the curve for isolated spheres. This indicates definitely that the increased effective size of the high potential electrode was a factor which cannot be neglected. Curve G, for the open cube at 6 diameters indicates an inorease in k over curve H, of' 9 to 12% between spacings or 40 to 100% of diameter. These values are of particular interest because the oage at 6 diameters roughly approximates the High Tension Laboratory enclosing the l meter spheres (6 diameters, for 1 meter spheres, is 20 feet). 24 PART III .Analytical Treatment ot the Effect of Grounded Bodies. From qualitative arguments, it is olear that the disturbing effect of all ground planes and shanks will beoome of greater importance as the spacing of the spheres is increased. Since other and quite satisfactory means are available for the measurement of voltages up to 300 KV• no great interest is attached to a knowledge of these disturbing effects below 20 cm. spacing for l meter spheres. On the other hand, the maximwn force for voltages just below flashover decreases rapidly for spacings larger than one diameter. The disturbing effects are therefore cal- culated for a spacing of one diameter, with the understanding that the results will be maximum values which will never be exceeded in practice. A further reason for choosing so large a spacing is that the infinite series of images converges rapidly and hence only a few terms need be considered. The effect of a single infinite ground plane, parallel to the line of centers of the two spheres and distant from it 5 diameters will be calculated, the in.finite series for the effect of four planes forming an infinitely long prism of square cross-section will be developed, and the effect of enclosing spherical cages of ten and tiventy times the diameter of the voltmeter spheres will be found. To make the method clear it will be applied first to the case of two isolated spheres (8) of unit radius at a spacing of 2(• 1 diameter). one sphere being at potential 1 and the other at potential zero. Force Between~ Isolated Spheres at!:_ Spacing~!_ Diameter. The basis on which the theory of imag~s rests is that if the real conductors are removed and an artificial system of charges is set up such a. That is, the radius will be taken as the unit of length. the spacing is 2, and the center-to-center distance is 4. Hence 25 that equipotential surfaces are created which exactly replace the real oonduotors, then the field outside the equipotentials will be identically the same as the field outside the conductors. It is necessary, then, to remove the two spheres and to replace them with a set of charges which will produce identical equipotential surfaces. A unit charge, placed at the center of the sphere at potential 1. will produce a spherical equipotential or unit potential at unit radius, but it produces an unwanted field at the position of the grounded sphere. It is easily shown that a spherical equipotential of zero value and radius a will be produoed by any charge eat radius rand a charge - e • a at Jr 2 radius!:. on the straight line connecting the center to charge e. r· Hence 2 a oharge - 1(-¼) (• .250) placed at a radius! (• .250) from the center of the grounded sphere will combine with the original charge to P."oduce an equipotential which exactly replaces the grounded sphere. But this charge has distorted the equipotential which replaced the sphere at unit potential. To oo?Tect this difficulty a third charge must be added• ,<• 1 whose value is - (- .250) + .0666?) at a radius {4 - .250 1 (• .2667). The addition of this charge has distorted the 4 • • 250 grounded sphere and hence another charge must be added inside it. A systematic tabulation will be necessary tor clearness. let the origi?J&l unit charge be la belled q0 , the first image be 4i-, its image be q2 ,, etc., and let the distance from the center of the nearest sphere to charge 4n be rn• Charge nun1ber The following tabulation results: Sphere at Unit Potential Sphere at Zero Potential Value or Value of 1!i. in 4n} C!19-rge 0 1.00000 Radius o.o l 2 .06667 .26667 3 4 .00478 .26794 s.1,9.11 s,s,10,12: .00037 1.07182 .26786 Radius Char~e -.215000 .asooo -.01786 .%'.6786 -.00138 .26795 -.2692'4 26 1 rn • (4 - r - 1:J nIt is seen that some or the charges have been combined. · This ha.s been done because they are so close together as to be indistinguishable in the computation of forces. To find the force between the spheres, it 1s only necessary to find the sum ot the forces between all the charges within one sphere and all the oharg~s within the ~ther. The attractive force between two charges is of oourije, minus their product divided by the square of the distance between them. Thus: • - qii•4m (4 •• ~~ • rm)fr For example, the force between·q9 and q3 is t- .. ....., (.06667)• -.017867} 4 - .26667 • .26786 •• • .000099 - The results or this and all similar calculations are tabulated: Force Charge Numbers O• .017'1"/8 1 .00128~ • 000099 2 • 1 .001374 2 - 3 .000099 2 - 5,7,9,U .oooooa 4 - l .000099 4 • 3 .000007 . ~ Tote.1 f'oroe • .020746, which may be rounded off to .02075 , In regard to units, if the potential or the one sphere is one stat-volt, and its radius is one centimeter, then q0 must be one stat-coulomb, and all forces are in dynes. It should further be noted that, if the 27 th radius of the spheres is C centimeters, then then charge beoomes ahd distance dn becomes cc3.n. And f . _ (Cqn)•(C4m) • - 4n4m 2 in dynes, just as before. (Cdmn) 2 (dnm) The force is thus independent of the radius of the spheres (as long as the relative spacing, and the potential is maintained). If' the potential is E stat-volts, charge q0 becomes Eq0 , and the force becomes E2 t. The force in dynes between two spheres at any spe.oing, s, one at potential E and the other at zero, becomes, then, where f 0 (S) is the force on two spheres at spacings, with one at unit potential and the other at zeroi that is, f 0 (S) is the value for spacing S corresponding to the figure .02075 which has just been calculated for spaoing 2. If the force is expressed as F grams, and the potential is V kilovolts, these changes in units will require this equation to be rewritten :,, J f(dyru,s)" F (980.6) • E"i'0 (s) • fo.!99 or V )• F (980.6)•(.2998) (kv · f (S) 0 V • 9.389 ,rtr ~ ·y F Or in V t 0 (s) 2 k • For the example chosen, isolated spheres at 1 diameter spacing, k 111 9.389 • .oao75 • 65.l KV. gm• a- The Effect 5>f One Infinite Grou~<! ~!_ane. The value of f 0 will now be oaloulated for the case of two spheres at l dia:meter spaoing with the addition or a ground plane parallel to the line of centers and 5 diameters distant £ram it. As before, let the radius of the spheres be the unit or length. Following the same argument, it the spheres and plane are removed ~ 28 and replaoed by the set of oharges found before, there will be equipotentials or the proper values to replaoe the spheres, but the plane will not be so replaoed. To produoe the desired plane equipotential, for each oharge in front of it, an equal and opposite charge must be placed behind it. Ee.oh such image charge must be on the extension of the perpendicular from the original oharge to the plane, and as far behind the plane as the original charge is in front of it. If' a set of oharges identical to those used to produce the spherical equipotentials except of opposite sign, is so plaoed behind the plane, then the plane beoomes an equipotential, but the spheres are now distorted. This second set must now be re-imaged inside each sphere, and eaeh new ohe.rge in one sphere must then be imaged in the other sphere. All these ohe.r..ges will have distorted the plane, and the whole process must be repeated. The computation is greatly simplified if advantage is taken of some approximations whioh introduoe only a very small error in the calculation of the foroe. The charges to be placed behind the plane are 20 units from the spheres (plane is at 5 dia. • 10 radii) and henoe the re-image of each of these charges within the sphere is at 2- 1 • •.05 units from the center. Sinoe, further.more, this distance is 20 nearly perpendicular to the line of centers, the effect on the other sphere of' an accurately placed charge will be indistinguishable fro1ll. ' the effect of a charge of equal value. but plaoed at the center. I£ all the re-images in eaoh sphere are to be placed at its center• then the charges behind the plane may as well be lumped together at onoe. One further approximation should be made. It is almost obvious that oharges whose positions differ only l or 2 in the third significant digit will have an effect indistinguishable from that of a single charge of their lumped value at their mean position. Thus in the table on page 25, oharges 2 and 4 should be lumped with 6• e, 10, 12; and 29 charge 3 with 5, 7, 9, 11. When these combinations are made, that table becomes: Charge number Sphere at Unit Potential Sphere at Zero Potential Value Value of Charg,!_ Radius -.25000 .250 -.01924 .268 Radius or Charge + 1.00000 0 l 2 • 12 3 • 11 + .ona2 o.o .267 + 1.01182 -.26924 The net oharge in the two spheres is+ 1.07182 - .26924 • +.80258. Henoe the lumped charge behind the plane will be -.80258, and the re... l image or this in eaoh sphere is - (-.80258) 20 • +.04013. It now appears necessary to repeat the oaloulations for the two isolated spheres, starting first With q0 • .04013 at the center of' the grounded sphere. If these caloulations are made, it will be disoovered that t.'lie net oharge on the two spheres is no longer .80258 but some larger value. Hence the value .04013 is not correct and another series of oaloulations must be carried thru. There is a short-out, however, by which much of this labor can be avoided. It consists in noting that the ef• teot of starting with a charge+ q0 at the center of' one sphere results finally (in the case of' isolated spheres) in a total charge on that sphere ot 107.2% of q 0 , and in a total charge on the other sphere of - 26.9% of q0 • Now let 40 • 1.0 and let e be the re-image charge (.04013 in the first approximation above). Superimposing charge a at the oenter of the unit potential sphere will add 1.072a to its final charge and will add -.269a to the charge on the grounded sphere. Similarly, placing charge a at the center or the grounded sphere will add+ I.072a to its charge, and -.269a to the charge of the other one. The total charge on the unit potential sphere is now l.072 + l.072a • .269a, &nd that on the grounded sphere is •e269 + 1.072a - .269a. 30 The net oharge on both spheres is (1.072 - .269)(1 + 2a). But a is 1 to be - (- 20) times this net oharge, io (l.072 - .269)(1 + 2a) a• • .04013 + .08026 ~04013 o: • .9197 • - 0436 Some detailed caloulations must be made to find the force, but they need be done only once and in these calculations the figures of the table on page 29 may be used as percentages so that muoh of the tedious part oan be avo,ided. The 11te.rt will be ma.de with 40 • (1 + c) • 1.0436, and all charges closer than .001 will be combined. Sphere at Ground Potential Sphere at Un:f:.~ Potentia~ Charge Number 0 Value Radius of Char;e Charge + 1.0436 o.o + .267 .0749 Radius ot l 2 to 12 3 to 11 Value -.2609 .2'50 -.0201 .268 o.o 0' l' 2' to 12' 3' to 11' - .0109 .250 .oooa .268 .267 + 1.1068 -.2343. The net charge is 1.1068 - .2343 • .8725; hence the image behind the plane should be ••8725 and the re-image within eaoh sphere(o:) should 1 be -(-.8725 20 ) • .0436, check. The forces may be calculated just as before, exoept that the foroes due to the charges behind the plane need to be considered new. For this purpose, the lumping of the oharges of opposite signs will not be allowable because the images or those in the grounded sphere produoe ·no component of force (on the grounded sphere) ' in the direction of the line or oenters. The desired component of the force due to the charges behind the plane is 31 f 1. - (-.2343)(-1.1068 ) • 4 42 + c20J 2 V4' + (20,2 • -.00012 A .further saving of computation time may be made by noting the.t eaoh of charges Oto 12 are inoreased over those in the table on page 29 by 4.36%; the total foroe between them must then be that found be2 tore, (.02075), times (1.0436). Similarly, oharges 0' to 12' are just 4.36% of these values and the forces due to them. are therefore (.0436) 2 ' Thus the only forces that need to be oal- times the original force. oulated by <ln4m ~ are the forces bet>veen the unprimed oharges and the Tanm, primed ones. The forces as tabulated below, are for convenience aeP- arated into attraotive (+) and repulsive(-). Charge numbers .Attractive (0 to 12) • ( 1 to 11) (0' to 12') - (1' to 11') l to 11 a.nd Foroe . Repulsive· .02269 .00004 .00012 - Image 0' to 12' 0 0 1 0 2 - - .00285 0' - {2' to 12 1 ) l' - {3' to 11') .00023 .00023 .ooooz 0' .00024 -.00369 + .02263 Net attractive force with ground plane .02263 - .ooss9 • .01894 dynes Net attractive toroe with spheres isolated • .02075 Net reduction of force due to the ground plane ~Effect~~ Ground Planes. The next problem is to find the force by the method of images when the two spheres, still at 1 diameter spacing, are enclosed by four ground planes parallel to the line of centers, torming a rectangular prism of square oross-seotion. This case is of some importanoe because real planes of finite width may be treated mathematically as it they were of infinite extent. The mathematical difficulties ot dealing with tho "edge effeots 0 of finite planes are thus avoided while at the same time the inaocuracies ordinarily acoO!!lpanying the ohange frOJll theory. involving infinite planes, to practice. involving finite ones, aro also avoided. It will be assumed, as in the previous oase, that the planes are at suffioient distanoe from the spheres that the sets of oharges in the two spheres may be combined into one in the first image behind each plane. If' a cross-aeotion thru the . center be taken, it is seen that the whole plane must be sprinkled with images to make the planes equipotentials. (See Figure 9, page 34.} If a is the distance between adjacent images (in terms or sphere radius as a unit) and+ q is the net charge on the two spheres, then the re-illlage within either sphere of any one of the first four images is • ( .q.;.)• q; • a· a Sinoe there is symmetry about both the horizontal and the vertical axis, the effect of &.11 the images will be 4 times the effect of all those in one quadrant. 'l'he lower right-hand quadrant will be used. The re-image of the change in row o, column n, is The sum of the re-images of all the oharges in row O is c,,CI Q = !L '\" (- I)"'"' 0 "'L n=, n The re-image or the charge in row 1. column n is ad the sum of the re-im.ageG of all the charges in raw 1 is 33 The sum of the re-images of all the oharg;es in row s is • -:, == Qs 1- , oc. (-Js+n-t-1 L Y. _5:1. -t nA- n =1 Combining the re-images for all the rows: = <: (-,J'•n+1 L L ~ Qs = ! ~ Q L rs1,+ n,.- s=-o S=o 11=1 It is plain that this series has diagonal symmetry with respeot to the two parameters sand n, by ta.king advantage of whioh the labor or oamputation oan be out almost in half'. That is, the term ·n • 55 1 s • 72 has identioally the same value as the term n • 72, s =- 55. It is only for charges on the diagonal (n • s) and in the first row (s • 0) that there are no similar ones with which they oan be paired. (9) Before this pairing is attempted, it should be noted that the double series is convergent. First, for rows the series Q8 can be compared term by term with the convergent series L~ C-!_m ,,. Form=- n = k Therefore the series Q is convergent. s Furthermore, and the sign of Q8 will be that of its first term. Similarly it can be shown that Q8 + 1 is convergent, and it is also true that -!- _j_ S+I I 9. The values in rows• O could be paired with the values in column n a O but if this were done, more than one quarter of the total number of charges would have been included. 34 FIGURE 9 @ Or/9in_a l n~t ,harq~ (90 ) 0 Posit, ve 1ma9es Ne9ative 1mo9es • sII n= N I 0 I ~--i • - 1---· - - • I 0 ;._ .3 4 0 • 0 -- · - - -- - - -----· - - . . .. ·- ·---- 0 0 -- - - - - t - - - 0 2.. • • 3 0 - - - -- • ~ - --- - • 0 0 + - -- -- · - - - - - • 0 • 0 • 0 - - ---- --- - - - - -- -- + - - - - • 0 • • 0 • 0 0 -- --- - -- --- - 0 4 • - • 0 -- 1--- --- - - -- - - - - 6 - -- -r------- - - - - - .J' 0 • 0 • 0 • 0 6 • 0 • 0 • 0 0 35 and that Q8 +~takes the sign of its first term. Now sinoe the sign of the first term of the several series oscillates. the a~ of the (-1) m sums oscillates, and the comparison with the series demonstrates the convergence of the series or the sums. To return to the sim.plifioation of oanputation by the use of the symmetry, let the rows• O be separated first: = .$.. { Q o<. ~ (-,Jn+1 + ~ ~ (-l)s ~n .,,} L L L n n=-1 !>=-I rs7-+n:z. 17=1 The first series is recognized as one whose sum is loge 2. Now let the terms tor whioh n • terms can then be paired. ' s be separated. _.,, PO _, The remaining S-117~/ DO } /-,z_ I- \ (-1)2-s-11 f- 2 ~ \ -(,.........__=!)-=--Q =!~ {/V'(7 L fzs& Lts~~n& L S := I .S: I 17 :: S~I This step has had the unfortunate result of introducing a divergent series, ,:;,o l oO (-I) ;1..S-I/ J/.2.s· s:1 = -~L - I 5 s =t Since the original sum is convergent, it is presumed that this means simply that the reamining double series is also divergent and of opposite By physical intuition. it seems safe to make the calculation tor a square array, that is, out ton• s • k rather than out tons+ s• • k• which would be a circular array. In any ease. since the double series is oscillating, it seems safe to say that the maximum error will be limited by the largest term negleoted. Now the first term, log 2 is .693 1 and it would be interest- ing to see how far the oaloulations must be carried before the indi• vidual tenns are as small as 1% of this value. Talm s • I: to find n such that 36 If the upper limit is finite.., the double series can be written ~ ~ L L S: I (-!}Sf,,,, Jrs~-1-na.. n::~+I in which the number of terms is N • k (k + 1) ~ If n must be carried to 144, then k + l • 144 and N • (l4S~(l44 ) • 10,296. -..s , ( In addition, there are 144 terms in the single series me.king a total of 10,340 terms. s=t 37 ~ Effect 2£!:. Spherical Enclosing Cage~ Dia.meter Ten Times That of the Vol-bneter Spheres. (Voltmeter Spheres at 1 Diameter Spacing.) The method of computation will be within each voltmeter sphere, 2) to calculate the image of each of these thru the enclosing oage, and assumed values. 1) to assume the total charge 3) to oheok the aaouracy of the This check is made by comparing the sum of the images of the other three charges into one voltmeter sphere with the total charge assumed in that sphere. It should be noted that a unit charge at the oenter in addition to the images is required to 1-a.ise the one sphere to unit potential. Let the radius of the voltmeter spheres again be taken as the unit of length. To simplify the computation, all the charges in the high :potential sphere will be assumed to be concentrated at its center and, for the present, all charges in the grounded sphere at .27 units from its center. (The figure .27 is the radius of the center of gravity of these charges for isolated spheres.) For the first trial, the values for isolated spheres will be used: (10) + 1.01 in the high potential sphere and - .21 in the grounded one. The images of these charges outside the cage, and their distances from its center a.re: •(-.27) 10 :a (lO) a 57e8 2 - .27 2 - .27 Under the influence of these four charges, the three spheres should be equipotentials. D + le56 at If they are, then the total charge within each should be the sum of the images of the other three. 10. Charges are in electrostatic units and fo1•oes are in dynes. Refer to page 26 for complete discussion. 38 The images within the high potential sphere are: - 5,;38 48.0 + 1.56 - 59.8 + .111 .. - .026 + .157 The unit oharge is Total 1.000 1.157 whioh is to be oompare4 with the assumed value of 1.010 The images within the grounded sphere are: - 5.35 • + .103 • __,,,......,,... 52.0 + 1.5s - 55.8 • - .028 Total - .193 whioh is to be compared with the assumed value of - .210 For the second trial, the charges assumed will be+ 1.16 and - .190. The images outside the cage then become - 5.8 at 50 and+ 1.10 at 57.a. The images within the high potential sphere are: • 5~8 - 48.0 • + .121 .19 3• 73 • + .os1 + 1.10 59.8 = - .018 + .154 The unit charge is Total Compared with 1.000 The images within the grounded sphere are: - 5.8 - s2.o • + .112 - + 1.16 4.0 -- .290 + 1.10 55.8 • - .020 - • .198 Total - .190 assumed. Com.pared with For the third trial assume charges+ 1.154 and• .198. Their images outside the cage are• 5.77 and+ 1.145. The image• within the high potential sphere are: + .120 + .053 - , .019 + .154 The unit charge is 1.000 Total Cam.pared with 1.154 1.154 assumed. The images within the grounded sphere are: + .111 • .288 - .021 - .198 Total Compared with - .198 assumed. In the computation of the force on the grounded sphere the accuracy will be improwd by separating the charge within it into two partas 1) - .288 at .21 fro,m its center and 2) + .111 - .021 • + .090 at the center. The foroe on eaoh of these due to the other three charges is: 288)[ 1.154 2 ( - - • ( • + • ( 3 . 7 3 ) - ( 5.77 2 5 1 . 73 ) 1.145 - 5 6 . 1 • •.7 090) [1.154 5.77 1.145 [ 4 1 - (52) 1 - (ss.a) stj t • .288 (.08294 - .00215 • .00036) • .090(.01212 • .00214 - .00037) • .01693 dynes. This figure is to be cam.pared to .02015 dynes for isolated spheres .02075 • .01593 • .184 • 18.4%. .02075 40 Conclusion: The presence of e. spherical g:rotinded cage of 10 times the diametei- of the spheres results in a reduction of force of l8.4%when the Tol'bneter spacing is 1 diameter. 41 ~ Effeot of.!:. Spherical Enclosing Cage~ Diameter Twenty Times ~ o f ~ Voltmeter Spheres. (Voltm,ter Spheres at 1 Diameter Spacing.) To make a similar oaloulation for an enclosing sphere of 20 times the diameter of the voltmeter spheres, assume charges of value midway between those for the 10 diameter enclosure and those for isolated spheres: asswne + 1.12 and - .230. The images outside the oage are 2 20 1.12 ) 20 ~ - 11.20 at ( 2~). = 200 - <• r (20) • 231 <- .23oh. 73 = + 2.66 at 1.73 2 The images within the high potential sphere are: • - 11.20 - + .0565 198.0 .23 3. 73 ., _...,...........,,. • + .061 ·, :a - .0114 + .1068 The unit charge is 1.0000 Total Compared with 1.101 1.120 assumed. The images within the grounded sphere aret - - 11~20 = + .0555 202.0 + 1.12 • -- • - .2800 4.0 + - 2.66 229.o = - .0116 Total Compared with - .2361 - .230 assumed. 42 For the seoond trial assume+ 1.107 and - .236. The images of these charges are• 11.07 and+ 2.73. The images within the high potential sphere are: + .0559 + .0633 - .0117 + .1075 The unit charge is 1.0000 Total 1.1075 Compared with 1.107 assumed. The images within the grounded sphere are: + .0548 - .2'768 - .0119 Total - .2339 Compared with - .2360 assumed. As in the previous case, the charge vrithin the groundem sphere will be divided into - .2768 at .z.7, and + .0548 - .0119 at the center. The forces are: l.107 f C = l' .C429 11.07 2.73 J • ·<•• 2768 ) [ \3.73):! - (201.1) 2 - (229e3}2j ·: -(+.042:9) (1 .1~7 42 r 11.01 2.73 ] - c2oa:),e-- <229) 2 = + .2768 (.07956 - .00021 - .ooos) - .0429(.06919 - .00021 - .oooos) a e2768 (.0792) • .C429 (.0689) c .01898 dynes • • 02075 - .01898 _ _ _.,.,,.,,..,._. _ _ • .oss = 8.5% .02075 Conol usion'-1 A grounded spherical cage of 20 times the voltmeter sphere diameter decreases the force a.sf~ 43 Compil ation of Results or Calculati ons ~ _!;he Ef:feot ~! Grounded Bodie~• 2' Since F • K • V , or V • k VF, the percentage reduction in the £oroe at a given voltage is approximately twice the increase in voltage tor a given foroe. For spheres at a spacing of l diemeter, the presence or the grounded body in column l results in the changes from values for isolated sphere stated in oolumns 2 and 3. (l} Grounded Body (3'.) Reduction in Foroe at a Given Voltage One infinite plane parallel to the centerline and 5 die.meters distant (3) Inorease in Voltage at a Given Force (Increase ink) Sphere of ten ti.mes the die.meter of the voltmeter spheres Sphere ot twenty·: times the diameter ot vol'bneter spheres The values in column (3) are values by which a voltage, obtained by toroe measur8l!lent and calculation on the assumption of isolation, must be increased to get the true value. 44 APPENDIX I Typical Data for the Determination of the Effect of Shank Size ~ Capacitance Measu~ment. The data for e. typioe.l capa.oitanoe measurement run is given on page 45 and a photograph of the original curve sheet plotted from these d&ta appears on page 46. The two curves marked "Capaoita.noe - Spacing" are plotted from the data given. to the same "Spacing'1 see.lee but to different "Ca.pe.oi tance" soales. The one on the right is plotted on the right-hand "Vernier" scale. and that on the left on the lefthand scale. The straight lines are those traced along the mirror as described on page 11. From their slopes. the slopes of the tangents to the curves were computed and plotted as the ourve marked .,Slope l" Spacing". This curve was combined with another for the same q '' ) shanks taken on a different day, the ordinates changed from Vernier ..! Divisions per Inoh to Kilovolt grams 2 (k) and the result plotted as curve c, Figure 4• page 14. 45 TYPICAL DAT.\ SHEET l" 4 Shanks Time 10:57 Stiaoing Touching 1485.4 1.2 1 • ., 2.2 1473.7 2• ., s.2 3.7 4.2 4.7 o.s l~l 1.:e 2.1 2.;s S.l 3.6 4.l 4.6 o.;ss 1.0:. 1.5 2.0 2.s 3~0 3.5 4~0 4.;5 s.o: o.;9 1.4 1.9 • 2.4 2.9 3.4 3.9 1468.95 1464.8 1462.65 1460.:ss 1459.5 1458.45 1467.5 1493.7 1475.15 1469.;65 1465.6 1462.;95 1461.;2 1459.7 1458.55 1457.6 1488.;45 1476.85 1470.;8 1466.;55 1463.6" 1461.55 1460.05 1458.85 1457.9 1457.;o 1478.9 1471.5 1469.f, 1464.l 1461.9 1460.i 1458.95 14ss.o 1457.15 1481.3 1472.5 Resistance 9880 9880 9880 9885 9885 9886 9885 9885 9886 9875 9875 9875 9875 9875 9875 9875 9875 9875 9876 9876 9875 9875 9875 9875 9875 9875 9895 9885 9885 9885 9885 9885 2.;3 2.a 3.3 3.8 4.3 1464.35 1462~1 1460.45 9880 1459.1 1458.05 9880 4.8 1457.15 9880 4.9 o.a l.31 1.;a o.s 1467.9 Touching Room Tem:21 11.a0 9880 9880 9885, 9885 9886 9886 9885 9886 9880 9880 4.;4 1h29 Vernier o.s o.;, 9880 r,.a 0 46 ; tt • I ► ' • Q ... HJNI N?d ~NOl~IAl(I Y~INYM - ~d07S .FIGURE 10 Working Graph ot Capaoitanoe Measurements. 47 APPENDIX II Calibration~ Transformer .!:z Sphere Ge.p Flashover. Since considerable confidence was placed in the validity of the model voltmeter measurements indicating rather exaot linearity between tertiary volte.ge •v• and secondary KV - v. tor the transformer up to 100 KV. and since this conclusion was at variance with the sphere-gap calibration curve run by Lewis in 1927• it was decided to re-run the sphere-gap curve. For this purpose• while the setup was still on the tower. the grounded sphere was clamped and flashover measurements were made. 3-" The hose resistor referred to was replaced by 25' of 4 garden hose thru which tap water was run continuously. From the nature or the setup. it was impossible to change the gap spe.oing while the voltage was being applied. read the scale on the shank• The procedure was 1) to 2) to bring up the voltage to about 90% or the expected flashover value. maintain it for at least 45 seconds• and then to raise it approximately 1% every 15 seconds. This process was repeated at least five times for every gap spacing. It was found that when the voltage was raised at a rapid rate. flashover occurred at a lower value than when it was raised slowly. The cause of this effect may be the "conditioning" of the gap as suggested by Peek. In any case. tor each spacing the first readings are erratic and are all below the later ones. These data are summarized in the data sheet on page 48 in which the last 5 values of tertiary coil voltage at each spacing are given. The flashover voltage for eaoh spacing was taken from. Peek's data. correction (amounting to 1.3%) being made for air density in accord with his recommendation. Figure 11• page 60. The results are plotted in F.aoh point on this curve is the average of the three tighest readings. Since the reactance of the transformer is so very low (1.24%) and since the current just before flashover must be 48 at a considerable leading power-faotor. no possible reason oan be sug• gested for the calibration curve to fall below the ratio line. It is therefore concluded that the sphere-gap test was run under test conditions essentially different from Peek'a and that results obtained by use of his data are not reliable in this case. 49 DATA SHEET FOR FIASHOVER TESTS Time Tertiary Vol ts Shank Read!n§ 7-s'§0 A.M. 14 Time at Flashover I 189 10:34 Shank Reading · 5 190 193 193 190 12 4 178 · ads 10 10155 3 42.a 42.5 42.5 42.4 42.5 11:08 10 163 162.5 159 160 161 161 161 8137 11 65 65 65 6_5 .3 178.5 176 179.5 179 85 85 84.6 85 10:47 7:56 Tertiary Vol·ts at Flashover 167 168 167 160.2 159 163 153 167 8:52 9 11:24 15 150.5 199.6 2.00 1s1.s 137 137 12,ss 13 137 9:20 6 105.3 105.8 106 106 106 9:30 4 65 65 65 65 10:20 7 124 123 123.2 123 200 199.5 i.35 12:54 Barometria Pressure Temperature Relative Humidity 188 187 187 187 Contact 742.3 llllle 21.s0 c. 76.0% 50 FIGURE 11 61 APPENDIX I II Typica.t ~ ~ ~ Determination ~ ~ Ef.feot 2!,_ Ground Planes ~ :Measurement 2!_ Force ~ Vol ta,g_e. The following table is a. typical set of data. taken with the 12.s cm. spheres as a force voltmeter. Fran column 1, the shank reading, and the reading at contact., 1.2cm,., the spaoing can be found. In oolumn 2, weight., the nominal value of the weight hung from the double thread support described on page 18 is tabulated. Columns 3 and 4 give the tertiary coil voltages as read on a 75 volt and a 150 volt voltmeter, respectively. These voltmeters contained spring pushbuttons so that they were never both connected to the oirouit at the same time. At 75 volts, the additional load of the 150 volt vol-µneter (connected as a special trial oase) reduced the reading of the 75 volt voltmeter from 75.o to 74.9. Column 5 gives the Cathetometer reading as read, and oolumn 6 gives the values of deflection obtained by subtracting the reading 4.so, for zero voltage, from each reading. The horizontal force due to the deflection determined by the cathetometer, oaloulated tram a calibration ourve, is tabulated in column 7. The horizontal toroe due to the weight noted in column 2, and calculated tram the trigonometry of the suspension triangle is given in oolumn a. ColU!Dll 9 is the sum of ool'Wllils 7 and a. and is the absoissa of the ourvea of Figure 12. page 53. divided by 1000. Column 10 is {Tertiary Coil Volts) squared and It is the ordinate or the curves on Figure 12. On this figure the set of data is plotted as the ourve next to the top one, that is, for Spacing• 99.1% of diameter. 2:46 0 6 0 13.6 2 Weight 1 Shank Reading . 4 5 NO CAGE 6 59.3 65.7 65.8 73.9 40.4 49.4 22.a , 31.7 0 4.ao 5.28 5.73 4.60 a4.a 91.2 97.4 0 4.17 3.71 5.84 6.26 s.s1 s.21 74.o 41.3 50.1 57.4 66.0 s.oo 4.60 4.81 .96 1.13 - .an .se .69 .12 .41 10.98 2.19 47.89 1.10 8.30 9.48 5.47 - .20 - ,43 1.24 1.62 1.91 .s2 1.00 1.65 2.45 3.28 4.32 ~ .13 1.so 1.so 8 9 Weight Total Force Foroe .24 .37 .56 .76 1.01 - .54 Foroe 7 .21 .40 .61 .91 1.24 1.66 - .89 Voltage Cathetometer 75 Soale 150 Soale Reading Def'leotion 3 -- TYPICAL DATA SHEET en N : . i' =-::._:--=.~ ~~ l -----1--· ,~ . ' ID · ~,-1. IN r CitllAMI .._ FIGURE 12 Working Graph of Force and "Voltage Measurements