The Design and Construction of an Harmonic Analyzer Citation Dumond, Jesse William Monroe (1916) The Design and Construction of an Harmonic Analyzer. Bachelor's thesis, California Institute of Technology. doi:10.7907/agz4-5r53. https://resolver.caltech.edu/CaltechTHESIS:07242024-225451767 Abstract No abstract. Item Type: Thesis (Bachelor's thesis) Subject Keywords: (Electrical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Electrical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1916 Record Number: CaltechTHESIS:07242024-225451767 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:07242024-225451767 DOI: 10.7907/agz4-5r53 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16590 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 25 Jul 2024 16:45 Last Modified: 25 Jul 2024 16:57 Thesis Files PDF - Final Version See Usage Policy. 22MB Repository Staff Only: item control page

i I I I I i ,. i . ! i I I I THESIS I I \_, n·esign and Construction o:f a:n. Harmonic Analyzero by Jesse WoM~ Du Mondo Class of Ninete ~n Hundred Sixteeno Department of Electrical Engineering~ THROOP COLLEGE 011 '.I.'ECEUOLOGY" I I I lr t! ! --.-- -··· ····1 Pasadena . California,. 19160 1 INTRODUCTORY. With the advent of the long distance high voltage power transmission line and the use of polyphase circuits the need for an accurate analysis of weve forms has become imperative. There is today no simple inex- pensive device on the market for mechanically accomplish~ ing an analysis of waves plotted t-0 rectangular coordinates. An Harmonic Analyzer is manufactured by· the West- inghouse Electric and 11anufe.cturing Company but is adapted only to the analysis of waves plotted to polar coordinates, and is moreover too expensive for the average laboratory.. Michelson ha~ patented a mactine which will both analize and. synthesize waves in rectangular coordinates but it is excessively complicated. and th~ price is prohibitive. Moreover while excellent in theory the operation of the machine depends on the addition of the forces exerted by helical springs so that its practical accuracy is dubious. In view of these facts tbe 1·01lowing design of an Harmonic .Analyzer adapted to analyze rectangular coora.inate curves just as the~ come out of tbe ordinary oscillograph without replotting cannot be considered out cf place. ·· In this paper under the title of "Tbeory", a discussion of the determination of the Coefi'icients of r: :E'ournier' s series v:ill be ei ven toget1jter wi tb a met'hod - - - -- - -----·- · -- .'/ . / ... t~\./-' · tw-r-· :·._r. . - ii::·< •. :.-;;-· . • 2 r;.i: ;._~ of representing ~r;f (e) ( sin n ( (.; cos n e ) ) d 9 -' e ) _graphically which as far as I know is quite {<nevi. /\. This •graphical method which I have named. the "Cylind.rical · Graph", waf:! a "by-product" in the _d.esig-n of the machine-. · ·~ -· : \ ' The problem was attacked. from the point of view _of a :}::;::> machi·b:- which . ?TOuld eval1~at~ the integral just me~tioned. ~:·;:-_: f:~t : {. •1::~·-· • In eA"J)laining the theory o:f operation of the Harmonic Analyzer it greatly simplifies matters however to reverse ·this order of events by first developing the cylindrical graph and then showing how the analyzer takes advantage of it. _ ... , .. . ' _ Uno.er the heading "Design" a complete. description : of the analyzer ~nd its mode of operation supplemented v:i th detail and assembl~1 cl.rs.wings will appea:r. The secticn entitled. "Operations" is intend.et to give concise directions for starting and using the Analyzer. Some space is also aevoted to a few simple tests ,, o:f curves which me.y be made by inspection and will save ~uch time in a complete analysis. At the date of this v:ri ting tbe Analyzer is still unt er cons-truction. If time will permit a fourtb sec- tion will be added to th is peper wben the .A!lo.l~~z er is com:;,letea gi vine the :resu1ts -of e. test of the machine. A number of patterns were ~ade of the perts of the i - - - - - - -- -- ··---- -------------- - ·-- ------- -·- .3 - --- ------------- .------ -- . - --- ·- --- .··· ·· -· -- - - ----- --·--- - ----- ---- .\'" \· .•J; - - - - - : - - - - - - - - - - - - _ _ ; _ _·-', .:....·.,_,___ I 3 inalyzer and these will be left in the Electrical depart- · ment of Throop College in case it should become necessary ~~ desirable to make duplicate parts. In the theoretical development of this machine con- stant reference was made to Eyerly' s text entj_ tJ.ed. • "F·oun\)ier' s Series and. Spherical Harmonics". ~N•:.. :-;- . In the construction of the m~chine I owe much to the ~ patience and skill of sevsrs.l of rny fellow students. Mr. l'rosser, Mr. Kramer, Mr. :Bui·ns, Mr. E.eywood. and :rlr. Wilcox. Without their help tbe machine would never have been completed. , I wish also to express my thanks to Dr. Sea.res of the 1:t. Wilson Solar Observatory who made it (.·, ' possible for me to have tee pla.nimeter divided on the dividing engine at the Observatory shops . .Above all I wish to thank !'rofessor Ford of Throop College for the kindly interest he has taken in both the woi·k of design and construction. His man~ suggestions and modifications have made the Harmonic .Analyzer a i::uccess when otherwise it would have been a failu..re.. •.· - - - - - - - - - - - - - - --- ----- ---- .:,,_., -- ---.. - -------~------· -- ··--···· · ···~-· ··•·- - 4 .TR"EORY. The discove:-y of the fa.ct that any single valued :finite function of a variable . can be expressed as a sum of sine anc. cosine functions of different frequencies of that variable waE: made by Four)\ier in 1822. It may· ··._ , .r.-_ - . ~ :J:r;• I , be ezpreseea. thus: ::::·.,-:,:·'.' ( 1) f ( el - A1 sine+ A2 sin 29 + As sin 30 + + * B1 cos e + E2 cos 29 - B~.,; cos 39 + ~ It is evident that the coefficients A1, A2 . ~ ~. 3 0 -. .. -~. - ._. .·"·: · f· •B1 , B2 . "' "'. he:rE. represent the arnpli_tudes of tbe harrnoI?,·1c components off(~). The presence ·o~ the con~tant B0 is harmonized ~ith the remainder of the expression by rega.rd.ine it as the coefficient of the term :B 0 cos 00 ~ Obviously A0 sin oe is omitted. The expression (1) is _perfectly general ~or an~ finite single valued function of e. Th'e~i function need. not even be continuous so long as it possesses. only a finite number of d.iscontiniuties. But the curie us ancl a.l togE-tn er :fortunate thir:G a.bout this expressio:r. is tho.t for any given iu.."1ction f (.X) the uniquely determined. The immense utility of' the Four'11,ier ~eries is entirely due to this fact. To una erste:nd t'his a little more clearJ.y see plete ·5 III. This shows some sireple sine curves of different periods and the resuJtant curve obtained by adding co- · rresponding ordinates. Tbe component of lowest fre~uency . is called the fundamental, that of tbree times tbe ire• quency the thirl harmonic. etc. Now the principle we wish to make clear · is that given any curv'e whatsoever • such as :f (9) in the figure III . there is one and.only one amplitude of funaamental which will go to make up this curve--.one and. only one ampli tud.e of second.. third or nth harmonic. ·.• .r Or to put it ner,ative1,. in repro- ducing the curve we cannot have less that th& proper amplitude of one harmonic and make up for this deficiency by having more tban the proper amplitude of another· harmonic. The form of the func.:tion a.lcne oete:r- • Dines irirnediately the ampli tuo.e of all the harmonic components. It is desirable then to find the . law by which the • · .•· amplitudes of the various hHrmonics d.eper..d. on the form \ of f { e) . I. • ( 1) f We "have the expression given ( e) = A1 sin e .I.-r A2 ein 29 +B 0 ... :r3icos e .J.. ? 2 cos 28 . ·A3 sin ~ E 3 38 t cos 38 + ,., ,.. * + ~sin ne ' * + Bncos ne. If ~e · multiply botb rne!T."bers by sin e cl e we obtain f (8) sin 8 a. e;:: A1 s1i'· e a.. e -r terme: of the form Ansin e sinned e + :S 0 sin e de t B1cos e sine de.:. terns :of the form B cos ~9 sin 8 de. n If we now inte grate bot~ sides bet~een the limits 0 - -.·-.,<-~ :__- _-.. -.__-__ -.. -_.- -_-_-. - - - - - - - - - ·- ·- - - - - - --- ------·---- ·- - .... _____-- - - - 6 and 2 7r we (II) obtain ·12-0 sin 2 e d 8. uf (8) sin 6 d 6 All the otb€r terms of the right hand. member red"u.ce to zero when integrs.ted between these limits as the following proof will show. '.r1o prove that : . 1 Terns o±" tbe form An . 2'ir .. sin 0 sinned e = 0 Integrating by parts· in e sin ne d e __n_c_o_s_·_n_e_s_•i_n_._·._e__s_i_n_·.·_n_e_c_·_o_s_e_ 8 j 1 - l'.12 . Between O ana. 2 'T'f this is evidently Q except in the case when n = 1. The fraction then becomes indetermin- .J.- ate (·o ) and ,~en be evaluated. 0 • gives A, rz r•l: 1 • Jo Thus the case of n - l sin 2 e de and is the only term in the sine series which rerr:ains. Let us see if an~' cosine teiT.s Tema.in. To prove that : Terms of the form En1 2 "' cos ne sin e a. e = O Intef-Tating by parts jcos ne sine d e . nsin ne sine - cos fie cos e n2 - l This evidently vanishes bet~een the limits O and 2'it I __J. except when n = 1. Tbe :fraction then becomes in- determinate and it is necess~ry to investigate it further. 2 The cese of n 1 gives the inte5rel 1 Bn1 "cos e sine de ~ 3ut cos e sine=} sin 29, } sin 28 d 6 = - } cos 29 ---- ·-·--·-··-- - --- - --- ---- --- - --- -- -- - - - -· j d . - - - - - - - - - - - : , ,•_ _ _______ _ - -----------------~ 7 This evidently vanishes between the limits e • 0 and e = 2ii • Thus it , is seen that equation (II) is valid, all ter~s having·vanished from the r~g~t hand member except the one shown. ·j sin e a. e . e 2 2 sin 29 4 :Between the limits e = O and e = 2 'i1 tt is evidently becomes-;;- and Equation II becomes. (2'ii )o • • f (8) sine de= A1 z 11 Solvir.g fo:r A, (III) = 1 • (0211 f 11 Jo (e) sine de . Here then is· the aesired relation between the coefficient A1 and f (8). Note that it is entirely inde- pendent o:f the other coefficients. in the series and depends for its magni tud.e c:ompletely on the form of i' ( 8). This is only one coefficient however. All the others includir.e those of the cosi~e terms must be found. To do this multiply both members of E~uation I by sinned e z,-hen n is tte nlL'Tiber o:f the sine term whose coefi'icie:::t - . is desired.. 9 Integrating both ~eD1bers betv1een t'he lioi ts = 0 and e = 21( as before, all the terms will vanish in the, right hand mc~ber except the nth term. 211 is ~---- ------L • -- -- -- 1 (211 f (9) sin nfl d e = A.. f •• I Jo The result 8 An = lr- [ {IV) 2,;'i' f ( e) sin ne a. e 11_;0 Ecre then is a. eeneral formula. for any coefficient of a sine term in the series. By multiplying both members of (I) by co~ ne de and. proceeding similarly we get: 1 Ir2~ :, En = - ( V) l •7.J'o f ( 0) cos ne d. 9 This formula is not exactly valid when n is zero as will be evident when the method. oi its derivetion is rethe called.. for i,mu.1 tiplying of both members of (I) by cos oe a.e 1 1 ::n:d:::::: i:o t:::t::::::eo:Yt::· fo:• 1~~:1: ::t O I' de • or of the form )o(2't'i' cos ne a.e, which is evidently ..zero except for the ter.n n = 0, where it becomes, (211 ' (211 3o)o de = 2-.rB 0 • thus Jo f (6) ae = + 2'i.E 0 • (VI) The Cylindrical Graph. We now have sufficient for~ulae to obtain all the coefficients : (IV) ---- ,- - ---- -- - --- - --- - --- - - - -- ·--··-- {0) sin ne a.e. 9 { V) l, (211 Bn - ~.:S-1 ''. ,.JO of l ( VI) f (9) cos ne d9. with the exception • (2'.: / f (8) d9 2'ii _;O It is well never to lose sight of the geo~etric interpretation of the problem as it effords a concrete conception for the ~ind to grasp and retain. Eow can we represent s:rabpically ,:_ " f 27i ( sin ne) ) d0? ( cos ne) f" ( e) ( o Let this be regarded as the integral of the product of a differential quantity lsi~~ne a.e) by some finite ~ quantity f ( e). Suppose we let the thing to be found is sin ne a.e = di\. • Then f f ( e) dt\ . Clearly this will ap:peer. es an area if we plot values of i\ . es abscissa ana f (9) as ordinates. I\ :But ·what is i\ ? ( sin ne d8 -.xcf" ---.._,_ ..I .?\ = - l cos n9 + c . 1 n 1 Consequently plot values of (- n cos ne + c) as abscissa end values of the f• (e) as ord.inates. Now if .• ttis is done betvrnen the L.::.::. ·. :.: 0 - 0 and 9 - 2..- . a curve will be ottained ·nhose area s'hould be the desired integral. enc. if we di via.e the ere a of this CD.TVe by JI we will .eet the value of An. Eere then is a simple ~ethod of obtai~inG graphically the coefficients in the Four~ier series for en::,,· function f ( 8). ··- - - - - - - ·-- - ---- -- l3y ruli~g paper - - -·---··------------- -- - -- ---·- ·-·- - - -- -- - ·- --· · · -- - - - -- - ·-- -- 10 with vertical lines so spaced that their distances . measured from an f (8) axis will be values of cos ne for equal simple increments of ne the work of plotting ca.n be, reduced, for it remains then only to assign to each vertical line i t.s proper height accord.ing to the value off (8) at tbat point and to connect the ends of these ordinates with a continuous smooth curve. If it is desired to find. the coefficient of a term in the cosine series similar reasoning will show that it is merely necessary to plot values of~ sin ne against n the given :f_ ( e) ancl the area divided by 1"r' will be the desired coefficient. Plate (I) shows an analysis of the curve f (8) = 1. 5 s:tn e by this rnethocl_. ,,.. . . Th_e unit for both abscissa and ordinates is one inch. 7ti th this unit the ordina:ry ,, planimeter will give the corr,~ct area directly. These / curves will be recognized as the familiar composition of two simple harmonic motions at right angles. In gen- eral. however, the curves are not restricted to this type as the form of the curve depends entirely on the form o:f:" f (e+. Since f (e} = 1.5 sine, if our method is correct we should obtain zero for all the cosine coefficients and also for all the sine c6efficients except A1 the coe f ficient in A1 sine. Obviously A1 stouln turn out to be 1.5. If the curve for Ai i~ ?late (I) be measured with a planimete:r.,: :i t -will be i'ound to --- ------ - - - - - - - -- . _ .. __.. .. . ________ ,, _- --.. · -- - · " ·- --- - .. -· ... .. _.. . . - ·- - - - - ·- -- - - - - -. ' I ~ ' -a: t~o,11e i i -: ~ ~. ~ , ·~ , · ., I::, · '<>' I I; I I I I I I I I I : I l'J=I n=I A,= u;;.o =/.:S" in. "A-!l!"e'1.- 0 11' - .I...), - ~ ~-....: I I : n!~ I 'I:> ' vil }I') \ ~~ "-I ' ~..i !I() ·' 1--L cos°'-10 ; i In i , i I I i n=z. B.,_ = O Flafe ([} n=-3 A.,-== O - - - - -·---------------- - - - - - - - - - - - - - - - - - 11 have e.n area of 1.5 x 1 x '7r • This is ob·v iously the case since the curve is an ellipse and its area is" times tbe product of the se~i-major by tbe semi-minor axes. Divid- ing this area by II we obtain 1. 5 for the coefficient as we should expect. Of course the same reasoning ap:plies to any amplitude r off (8) = r sine for 1.5. just as well as Notice moreover that _ in tracing this curve from e = 0 toe =Z,dthat is, from (- l cos e) = -1 to n (- 1 cos 8) = -1). the point moves clockwise. n If f (6) had been -1. 5 sin e the point v:ou.ld have traced arou...Y1d the curve in the counter-clockwise direction, as a trial will quickly show. The area would then have been negative and we should and would obtain -1.5 for the coefficient. --- As for the coefficieY.:ts of all the other terms of the series it will be seen that as the point moves from e - 0 to e = z" it traces either al'Jound no area at all or else around equal positive and negative areas so that the result is zero. I This again is what we should expect. , To visualize the who:!.e t1atter definitely suppose that the given f (8) which is to be analized into harmonic conponents . is plotted against eon a sheet of transparent paper. and that this sheet is then rolled into e right circular cylinder so that the ordinates become elements of the cylindrical surface and the point 9 ~ O coi~cites ~ith the point 9 = Tbe p:cojections of this curve upon a pla~e parall el to the axis of the cylinQer or its ·1; . l , ·:, • . , I ___,.___~7----------- J __ _ _ _ __ - - - - ----- - - - 12 appearance viewed at infinite distance would then be obviously the same as the curve for .A 1_ or ::c. 1 in Plate {I). Plate (II) is an attempt to show this in perepective. If the sheet were rolled twice before biineing e = 0 and 8 = 2 -u- into co inc id enc e the curve would appear like those in ?late (I) nhich represent Az or :s 2 • Here then at last we have a concrete idea of "?le .tin ~ r2--- 1 ., f I' f ( i e) (e) ( s n ) de . Bn :;-;: ) o ( cos e) can now look at curves to be analyzed with "eyes that see", for it is merely a matter of visualizing the . curve rolled up and remembering tbat the apparent area, which is seen is a !:'.)ea.sure ot the amplitude of · the curve's nth harmonic if the curve has been rolled n times. (Note the cyli:1der should be vie·wed. so that e = 0 appears on the eztreme ~ left for the si~c series and directly in the center for the cosine series). We have so far purposely avoided the constant term in the cosine series because it does not harmonize well v.ith the rest. Let us make both our im~gination and our transparent sheet of paper very elastic. Then if we stretch the c~rve f (8) on the sheet once around an immense cylinder of infinite cli2.!!leter brineing e = 0 and e = 2-;. togehter et the center as before, the apparent , ~s curve will ce::?:"tainlv ce f r l I I ~l . ." .." . ___ - _ ~ • .~a_,,,_ -0 (! a cos ae), (f (e)), and the area a..1~:id.ec.. by ·-:: S!lould be the constent term. Thie would be quite convenient if ~it were not that (:t) l... ;r:· ~ C'\! -~ ' ~ ~~ -~ ~ ~· ,V°) -\--. "i-.;:: ~ ~ O") ....... ~ II -~ ~ ~ t:::: ~ § II ~ ~ ~ - -~ ~ .<r;J- t:: ~ ~ G 0:2 ~ ""' ~ ~ ~ -- "'' - -....... ~ 13 while the point in treeing from e = O· .to e £ 2 'iT moves a.round. an infinite distance iJ,, invariably encloses what . would be a zero area if the distance were anything but inf·ini te. For, whatever the curve b~, if it is stretched out to in:fini te length any picture we can show of it Tiill be a straight line parallel to the stretched axis. Con- sequently .any attempt to measure the area will invariably result in an inde-termina.te of the form o0 - oo . 111oreover we cannot view the cylinder fro~ any position which will help us to evaluate this difference. ~he writer realizes the inadequacy of this explanation. The only way to coc~ prehend the matter is for the reader himself to experi\ ~ent ~itt some sarr.ple curves (known to contain a constant term) on this infinite cylin~er. The problem is fascin- ating but useless. Fortunately the constant term is obtained very simply if one gives up the idea of its being the coefficient of cos ( 0) 8 • ■- Thus a.s has alreac.y been sho.m : B0 7 .::. (211 f ( e) de '2:i _) 0 or in other words B 0 is the average ~ordinate. This can tesily be obt~ined from the curve directly Tiith a planimeter. In order to further ma~e cleay _the analysis of cu:rvcs into harmor.ic :fu.ncticns we will now take a curve apart and ... J:)Uv •r • J..• ....v ,. t oge J..b v ,er &.gc:.1n b O v!"l al~ebraica.lly and graphically. J..S Vic will st2.rt with tl-:e curve f (9) = e. a sim:ple _straight ~ ,,.-- -. - - - - - - - - - - - - - - - - - - - - --- - - - - - - - - - - - - - - - - ~ = 14 line passing tbru zero, and &nalyze it into harmonic components by the formulae IV. V. and VI. ·we will then take a finite number of these components and adding then together obtain an e.p:proximatiton to f (e) = e. Then, taking this approximation we will, by the cylindrical graph, resolve it into its original componer.ts again. If f (e) = e. sin e de = l i, • A 1 .= And 1 f )o 11 2 e sin e ae. 2tt "fi- ( sin e - ec os e) 0 = -2. in general: ~ : l/ f 'ii 211 Jo i\ 0) sin ne dB : ;")o :_ r2 11 e sin ne de 2 n . \ .,-, Let us see if f (8) = e contains cosine terms E1 = ! 1211 II 0 .r, .I. ( e) cos e dB = r2'i"i 1) " (cos e - e sin :Bl = .! r\\ e cos e a.e· 0 el 2TT = o. 0 And in general : }3 1 = - n . (2 ':i t G jo (2'i'i ~- > :. f ( e) cos ne dB = l ; ~Jo _ l ( cos n9 - ne sin ne) 2 ,.-:;- ::: o /·\1 -~ n 0 e cos n9 dB. ~ ,,.r- ·-·----------------=--:.:..,.,...---·----- ·- - - 15 ez-cept when n = 0 in which case Eviiently tbere are no cosine terms except 3 0 which is -- .. ;, The following then is the expression for equal to " f (e+ = e. sin e - . sin 28 _ f (el = e,., * -•-2s n i ne - "" _g__ 3 s-in 38 _,;. - * On ?late (III) will be found n the graphs of f 0 (0) - 'i'i". f 1 (e) =-2 sine. f 2 (9) =-sin 2e. These have been graphically added together on the curve sheet with the result shown at A. It will be seen that the resultant already ·with only the first four terms of the i •. infinite series approaches the ideal curve B quite ~early. e is plotted in radians with a scale of one inch equals one radian and f (8) is plotted to units of one-calf inch each. lTow let us start Yli th the curve A in Plate III and by the cylindrical graph analyze it into the same components out of which we constructed it. Plate IV. The wor:k is shov;n on For each value of ewe plot a point whose or- dinate is the ordinate of curve A for that value of e and whose absci~sa is - ! cos ne. n It is evident from an in- spection of t h ese fi gures ttet the tracing point has moved counter-clockwise around. the ~a jo:r part of all of the .I n=1 A, _=a';?'= -z.. ...:: n=2.. L1 - £!::.ti!.. - -/ n2.- rr - ----:: . ·, i ------------,----------''-'------------------------- 1_ . . . . - - - ; 16 areas up to the graph where n = 5. These areas tben are . negative and when divided by s-. are equal to the coefficients of the first four sine terms. Tbe graph for n • 5 however registers a zero area with a planimeter for the closed areas traversed in a clock-wise direction ezactly equal those traversed in a counter-clockwise direction. The graph for the fundamental cosine term has also been plotted and appears in tbe lower right hand corner. It is of course zero consisting as it does of equal positive and negative areas. Thus ~e have analyzed the curve back into the components from which it was built, the graphs ---- r showing conclusively that the curve is made up of four negative sine terms and no cosine terms. As for the con- stant te:rrJ cf the cosine series, its value is evidently 1\ ._,,, since that is the average ora.inate of tbe curve as can be seen by a glance at Plate III from the syrri.r-:etry of the figure. Tb e seal e in these :figures is unity equals one inch, both for the (- 1 cos ne) axis and the f (8) axis. n As an ~dditional exarnple the ideal curve f (8) = e shown at B in ?late III h8. s been analyzed for its first tv,o harmonics "by the cylindrical graph !:letbod. : The work is shown in ?la t e V. Tbe scaceof this gra.pb like the others is unity equals one inch. Since the graph is not a closea curve attention ~ust be paid to tbe area d~sired. To go "back to fun a.o!!":er:tnJ.s it v;v.s shown that, - ,~ -I, •.·, _ ,....... .... ~ ' ...1 17 An = ~ TT . wl:e:rc A - - · 1 cos ne = the abscissa of our graph. T~is n integral is obviously · the area . under the curve as v:e pass • :from Oto 2':'i, areas passed over in the positivedirecticn being p~sitive and vice-versa. The area desired is evi- dently then in this case by s-ubtraction the area betv1een tbe curve ABC and the ordinate AC. This graph is ob- viously a sine curve being the projection of an inclined line wrapped around a cylinder. - Since this sine curve has an amplitude of 1 anc, a period of 2 ~ , the area de- sired is evidently -2.; : dividing this by T. we get :ror the value of .e.. 1 , -2, which is v.•ba.t we obtainer. by the method of direct calculus. gre.ph divided by ~ Sir,ilarly the s.rea of the for n = 2 is -1; and a little co~sid- eration of the metbod. of construction of tbe figure ~ill ·, sho?, that for any graph whe!"e n = n the area divided _by 2 Ii will be -- which is what it should be. The area n traced over can be thought of as closed if vie consicler that the tracing point finally passes dov.~ the ordinate from C to tbe starting point at A. This corresponds to the vertical pa.rt of the c~rve shown in Plate III. If tbis c'i.evice o:f closir..e ir. t'he curve on the vertice.l ordinate is ~lways strictly observed with the planimeter tbe sign of the a.rea v1ill au.to!!:etically te.ke care of its elf and we thu.s have a ~crfectly ee~eral ~ettod of grephical analysis cepable of ~eeting all problems. f r --~ ... ___ ~- - - -- = 2., -1Je . f ( e) a. 8 .. 0 . l - - - -- - · ----· --·--- - - - - .. ... . C H I G --~ ,. ....._ ~ ('_ F 13 ~ ~ ' ~ · -.;:; r,'. 'k A ~ ~ -...:I I · ~\ +- t\l ~ ~ ti~ ~ ~ (I) -~ ~ "<::> t-.. . Ir, . ' n=J ~ ") ~ - A : : : _fl.!:£{L = -z. I .TT \Q ~ .IJ <::, / Qt<) );) <::::)--casne ,;::;.!? I() C) ~ IT\" ' - ~ ~ t'\I ~ I<) <;;;) • ' n ~ "t-t\J • ~~ l()O '-¼-,t) ~ -~ -~ a C) \ ~ t--i~ It) q le:~. 0in radians-:~ ~ "".... .... K)<\I "" n=?... ?/afe{I.) II. _ oreo - _ I 2-rr=-.- 18 Having seen the curve on ?late III the reader will rrobably ask why, if it is intended to reproduce the ideal curve E, it does not continue on to infir.ity as f (8) : e does. At 2,,.- it. drops off vertically to zero. The form of the curve depend8 entirely on the amplitudes A1, A2, A3 etc. If at any point on the curve these amplitudes cave not the proper value the ide&l curve will not be reproduced. Now the value of these arrplitudes it will be remembered was obteined by solving an equation. (See equation II above) But this equation possesses the curious property that it is only an equality between e - 0 and e - 2~ fo~ if some other limits had been used. the U."i.desirable terms o:f the form j t sin me sin ne a.e ~ould not bave vanished and Zquation II would not have been true as it stands. Therefore v/hen e becomes greater tban 2 s- t'te :formula. IV does not give the correct value for An and consequently the curve depending as it does on tbe An's cannot even approximate the ideal f (8). sc8.le however 2;; Bye proper choice or chenge of cs.n be marl.e to come an:,"V/here on tbe f (8) and thus ~e c an represe~t es much of it by the Fourier's series as we like, provided we are satisfied with a finite c.mcunt. i'!e v.ill now pass en to tbe design of a machine wh i c'h "Nill e.utor.,c. t ically d ete,rrr:ine t'h e intE:gral : ) -:: i J / 0 { sin :ne) .J ( 9) ( (cos ne) ae. e;iven the f ( e) . 19 DESIGN . . • In the study of alternating current wave forms the v.ave is now almost invariably obtained with an oscillOgraph. This instrument causes a fine point of light to move transversely on a very sensitive photographic film in t:.nison vii th the variation of the instantanious va.lu.e of tbe cur:rE<nt wave while at t'be same time the tilm has a uniform longitudinal motion. The period in inches of I the plot tbus formed may have any si~e whatever with~n the practical limits of the oscillograph. For the sake of simplicity in the resulting Fourier seriee it is importa.."1t that the units of the curve shall be so assumed as to ~ake the period equal to 211. To avoid replotting tbe curve to a new scale it is desirable then to have tEe I:'lachine devised so it will multiply o:rd.ir..atce whatever length the curve may happen to have by tbe proper factor to make it 2 ~. In the oscillographic record of the curve the value of the scale in amperes per inch is U.."1kLown. A calib- r~tion once for &.11 is impossible as the oscillograph re~uires contin~al adjust~ent. In order then to ascertain the sea.le to v1hich the oscillogrepb bas plotted. the w~ve it is custo~ary to usE a voltm8ter or ammeter at the time ~be greph o:f the -.,,,c._vc: is taken, thus cbteinir.g the root mean s~uare value of ell the instantaneous v1::.lues of current or vol~aGe grapg over a peri~d 2 \< n. It is rlI ./ _. , .,-· ~ .. . . 20 . desira.ble then that the instrU!.",ent shall be capable of obtaining this root mean squa~e ordinate in inches. The ampere or ~olt reading, effective value may then be divided by this R.M.S. ordinate thus giving the scele of the curve in amperes or volts.per inch. Then having ob- tained the values of the amplitudes of the harmonics in inches their values in amperes can .readily be obtained. · j;:---· A third requirenent of the n::achine whir.c is most imper- , ative of all is that its expense be small--a total cost of 'n-e75 bein~- the limit. , t -- "i'!i th these things in view the rnachir.e was aesigned. as follows . . Referring to the assemb1y drawir.g,A is a tablE sliding upcn a pair of rails BB. Projecting under this table is a large pin C ca~able of turning without lost moticn ancl ~aving a hole in its enlarged head to t&ke the roQ D, a good sliding fit. This rod D also passes through the pin E ano. through a third and similar pin O projecting U..'YJ.der the carriage F. wl:icb slid.es ·' on the rods GG. Thus the carriage F and the table A have a U..'Y1ifoi-r:i motion along their guides with respect to each otber. the re1ative rnag!!i tud.e of which can be varied by fixing the csrriege of tr..e pin E anywhere alone its eu:des The carria ge Fis fitted withe half nut ~eehes with the sere~ J. which The end Z of the screw J is adapteo to take chs.nge gear$ v:h ich trc.ns;;1i t motion be- I f _____________ ____....:...:___ _ ______ ~ 21 tween the shafts J and L. On the sh~ft Lis a worm M which drives the gear N thus turning the disc and quill Q on the stationary stud R. The disc Q has a hole bored exactly one inch fror.i the center which contains a pin S with a diametral hole through its enlarged head. The al'U.!!linum table T slides upon the rods UU and carries a rod. V which passes through the . dia.metral hole in S. . As by Q is driven round uniformly the worm M. it is evident II tbat the table Twill ex ecute an harrncnic motion of amplitude one inch. At the same time the carriage F will translate uniformly causing a proportigna.1 motion of the table A. · The columns WW are fitted with sma.11 V-e;rooved. pulleys in ·which rests the long rod X. A planimeter v rE=sts wi tr: its wheel on the table T, its stationary point being fi~ed in the outrigger of that table. The tracing point? of the planimeter is attached to the long rod X arid partir'ii'e s of eny motion which X may have. A po~nter Z on this sa me roa comes within a short dista.nee of the top of the table A. If a curve to be anel y zed is mou.ntea on this table the pointe:r Z can be made to follow the f (8) o.rdi:r:a..te of this curve e.s the curve slides along under tee pointe:r. !his mes.n s that the treeing point of the planirr:eter P is exectuing the same motion. What fi gure would this tracing point scribe en the aluminum table T as that teble mave s u..~der The ordin 2-te of the curve is evidently f (9) ~nd ( i t - - - - - -') -• . . .- ) -- - -- - - -- - ··- ------ -----·-- ··--- ·-·-- - ··- ···-·-· · . ·-· - ·····-- ·- - • -· \ ------------22 its abscissa is either of the for= a sin b8 or a cos b8 is . rrhere ft( an-a.- b ~ a et ermined ~-s-p-e·e·=H·v·e-3:y by the ±:n-.i:t-i-ed position of the point E and the gear ratio connecting Z and 1. Consequently the figure traced by P upon tbe table Tis the 11 cylindrice.l gra:ph" of the given curve on table A. It is evident that tb e a.re a of this graph is measured by the planimeter. If the planimeter is properly proportioned the reading on the wheel can be made to give tte traced area d-:tvided by~ I and thus the reeding on the wheel would give t'te amplitude of the desired harmonic directly were it not for·a circur:stance shortly ' to be explained. We mdst first give consideration to the units to wbich the cyJinc.ricel g:raph is tracec.. The wheel N has 72 teeth and since Mis a single threaded worm 72 :revoJ.uticns of L cause one revolution of :a. The screw J is cut vdtJ: twelve threads per inch. Thus to·~ove the carriage F six inches, 72 revolutions of Kare necesrary. ~ith certain rare exceptions the travel of the carriage Fis el~ays six inches for one ccmplete integ:ration from O to 2 ~ Now if the gear ratio bet~1een K anrl L is o:ne to one. t'he crank disc Q. will make one complete :revolution during tte no:rr.:.al six inv~ travel o:f tte ce.r:riege, F. If' the gear :ratio is such that L mekes 2 :revolutions for es.ob one made by .K tben obviously the cr~nk diec ~iJl ~eke two complete revolutions during tbe trave 1 of tr: e c2rr_iage F. '.r'he 1:::acbine LJ _ ,, . _ - - -- - .: ,-, --- 1. ! I I 23 is provided with ge~rs wbicb by ringing various ctanges ·will give all the integral ratios from one to sixteen, inclusive. Tr.e initial position of the carriage Fis . ma:::-ked upon one of the guide :rods G. The shaft K has at the opposite end f:rom the gc:.ars a crank provided ,:i tt a hole through which a pin can ·be inserted into the bracket "t>ehind. ';,'hiJe making the set-up preliminary to an analysis the :positio:r. of the she.ft K is al,l'ays fixed 1:,y this means. Ii the carriage Fis put approximately on the mark anu the talf nut dropped into place, tee carriage will take the proper initial position with great accuracy. This position is determined by means of the adjustable thrust collars on shaft K so that the swinging rod D shall be exactly parallel to its guides .HE. at the start . .> This prevents tbe table A fro~ changing its initial poEition ~hen the position of the point Eis ~haneed. Raving select ea. the positions on the curve that are ti \} It to be .tte I zero ) and I 2 ~ ) points, !!lou.nt the curv_e on the table A with thur.ib tecks so that the zero line shall be parallel v1i th tr..e _guic\.es along which the- teble is to move. -.,. The small marks at the ends of table A facilitate tbie. --r:/i "th tbe carriage F in the initial position see that the zero point o:f the curve to be analyzea. is directly under the tracing point Z. Unlock the half nut on the eerriege F and move ~the latter to.its extreme position six inches along the fUide rodr. The half nut ~ill locate the carr- -- - - - - - - - 24 iage accurately v:hen it drop.fl into the tr..reads. Adjust the pivot E in such a position that the point 2'j"; on tr..e curve ·;;-ill cor.:ie directly trncler the tracing point z. As above stated this will not affect the adjustment cl· ready ~e.de for the iilitial ::position. \'7i th its set sc:cew. Cla!!!p the pivot E ~7ben the tracing point Z 'has covered tb e di stance zero to 2 ~ of t1:e CUYV€ will have moved its normal six inches. tr. e carriP. ge :F This will have produced eza.ctly one complete revolution of tbe crank disc Q if the gear ratio between Kand Lis one, or n rev~lutions if that ratio is n. It is evident that the crank disc paeses through equal angles as the table A travels over equal distances. If the posi ti~n of' the crank disc Q is initially ae sho~n in the assembly dr&.:wing. the pin S as it moves around counter-clockv:i se will be displaced from t'he e.xis of tte roo.X. a distance in inches equal tc the sine of ne, n being tbe gear ratio and e the abscissa of the given curve on table A~ because the radius in inches of the circle described by S is exactly one. The teble T reciprccates ~itb tbc pin S. I£ a point had been marked on an eztension of this table directly under tbe pin J? on the rod .X while tbe table v1as in its initial :position, then at any i:r:stant the distance of P from t~at point ~ou26 be Bin ne, positive dietences btin g measured.on the table T from this point in the direction r.iarkcd o:, ".;"::e s:r.8.ll ar-::ow in tr:e c.rawjng and neeative dist f-nces in tee opposite direction. 25 Thus · i·t is evident that if the point P on the rod X be considerea. as tracing e curve on an extension of table Tit will et all times be sin n9 incbes from the above mentioned initial Y!:ark in the direction o:f the arro\'.1 anc f (9) i~cr.es fro~ this same initial point in the direction in which the long rod X slides, the positive directions here also1 being indicated by ai arrow. Thus it wiJl trace a cur-ve whose area in square inches \',•ill be n times as great es the area of the cylindrical grapb . . explained uncter ,iTheory''. It v.rill be n times af great tee.a.use tr.e "cyli:r.d.rical graph" was plotted between the coordinates! sin ne end. f (8) while here the curve is n plotted between sin ne and f (8). Consequently ii we wish to obtain in inches the amplitude of the nth cosine herconic it will ne necessary to divide the area of this curve by II n. :Now it is evident that the e.rea of this graph is ?:Jeasu.recl by tb e planimet er ·as the curve is traced, for the tracing point o{ the planimeter has followed the graph witn respect to the te.ble on v;hich the :planimeter lies. Tbe dia:!leter ot the z:teel o:f the planimeter ana. the length of its tracer arm ar~ such·that the area in sou.a.re inc'hcs d.ivided by-;;- is the sea.le rea.d.il'."...g. Thus it is nc:cessary only to di ·.ricJ e the area obtained on/ch e planimeter by n t~e nu~oer of the harmonic for which the curve is ceing analyzed, to obtain the cesired amplituce of that tar~on~a in inches. If tte initial position of the pin S r.ae been <JOO clockwise :f'rom that specified 26 above a little consic..ere.tion v:ill me.ke it clear that the abscissa of tbe graph thus traced would have b~en - cos ne. the The area thus obtained divided by n is then a□plitude of the sine term in the curve. Both of these initial points are plainly carked on the crank disc and are · located with greet accuracy when the initial set~up is mate by fixing the position of the shaft L. . This shaft. like K, has a crar.z o-ri one encl with a hole to permit sliding a :pin through iJto tbe bracket behind.. now that means have been provided for obtaining the sine and cosine terms, the next step is to develop a method of adapting the instrU!!1ent to find root mean square. values of curves. In "Alter!}ating Current ?benomena" by C.P.Steinmetz, the following method of obtainiJg graphically the R.~.s. value of any curve is developed. Suppose the given curve be plotted in polar coor&inates with f (e) as the radius vector and e as the argU!!lent. The small figure on the right shows a differential triangle on this plot. It is evident that the area of this triangle is ds = f (8)2 d8 a~a the total area of the -cur~e between the limits 0 and 2 ,;-;- is 27 2~ ' els= 0 root mean square ordi~ate of any function f ( e) is R • .M.S. - 1; 1 ( 2 :-: ,.1 I f ! 9~ r ; -d : 0 ~ (8)2 d8. ..,/ Comparing the last two equatior.s;I'Gt evident that (_ Tbus it. is evident tblrt""' i:f the given curve f ( e) be plotted. in polar coordinates it is merely necessary to divide the area by I\ to obtain the mean se;uare 'toalue and the root mean square value is of course the square root of this. In order to obtain R.It.S. values with the Ea~onic Analyzer it is merely necessary to lit¥ the reciprocating table T off its guides UU taking with it the pin S which I I slides easily out of the hole in the crank disc Q. Be- fore doing this the long rod X and tbe planimeter are removed.. The irregular shaped. al1l!7linum casting shovm on the right in the assembly drawing is then fitted onto the top of the crank disc Q, and secured with the thumb screws~ The long roa X anf the planirneter are then replaced with the r,le.nimeter wheel resting on the table of the aluminum casting. The pole of the plani!net er swings on a pivot on the outrigger of thie casting. The tracing point of the planirneter is pi~otea as before in the rod I. Bow, if tbc gear ratio· between shafts Kand Lis made one to one i~ is evident that the alu..~inum casting .. . .. · · - -·- - - - ------ ! 1 28 will turn once around while the curve on table A moves from O to 2" carrying with it the ,plani~eter. If. meanv,hile the tracing point Z has been guided alone the given curve it shoula be evident thet with respect to the aluminum casting the point ? has ·traced. the f { 8) plotted in polar coordinates ant that the readings on the planimeter give the area of tbat plot divided by ,-... I\ • If ·a pencil had been secured at P so as torr.ark on a piece of paper fastened to the alu,:1inum casting and rote.tine ~ith it. this polar plot would act~ally have been Since the planimeter divides the area in inches drawn. by -;;- automatically it is evident that tbe reading on the planimeter gives directly in square inches the mean square ordinate of the c~::-ve f (e). To obtain the root mean square extract the se;uez-e ::::-oot of the planimeter rcad.ine. In order to allow the oper£tor to concentrete his whole attention on following the curve correctly ·wi tb • the tracing point the driving o= tbe ~acr.ine is accomplish~d with a motor geared to tte sca=t L. brass bevel eee.rs 2.re user~. 'I'he ~o small !l1oto:r is e.d.a:pted. to run a1rectly OD eltern~ting current at 110 volts so it is m0rely neceesery to plug the ccrd connection of the motor t o an orQ' i ns.r:,1 e 1 e c vr1c .L.. l ,.,_v _ '1gr: soc.ire ',L. 1, • .' The har~onic analyzer haE but few adjust~ents. In ,,________ 1-· ./_.,_ ' -- !I 29 l I fact. the only adjust~ents aside from those incident to each set-up are the settings of the tbrust collars on the shafts Z: and L. The entire accur~cy of the enal- yzer depends on these collars however so that care should be taken not to disturb them. It is by means of the collars on 1 that the crank disc is finally adjusted accurately for its initiel sine or cosine position. I A good method of locating the initial point is to follow the process cor:imonly used. in locating the "dead centers" · of a steam engine. t'!ith the table Tin pl2ce on its fuides and the thrust collars on Lin any convenient setting, turn L by means of the crank until the table T· is approximately 90° from its ndeaa. center" and lock the crank. Scribe a light mark somewhere on the table T at right angles to its line of motion e.na set up a rricrorreter microscope above this mark so that i~ is \ exactly under the cross hair. Now unlock the crank ·ar.d turn 1 tmtil the table T having gone once iorward and back bas brought tbe mark a;ain U...."1d.er the mic:ro·met er microscope. Count t'he turns. Unless the collars on 1 are already in adj~stmcnt the crank Tiill not be in positioc to lock. Turn it 8.TOlliid hov:ever until it is in :posit:.o:r. e.nd loc}:. The cra..'1.k should tave made an o~:en nu.r,1ter of revolu.tior.s. If it has not. make an extra revolution be~ors loc]dng. The mar~ will now not appear under the cross heirs o! the ~icro~eter ~icroscope . •',.1..., .,~ ... h th.e :nJ.crorDe.,er • ... screw me1:1su:?:"e tbe clistance of the -----... - - - - - - - - - - - - - - - - - - ! 30 separation end divide it by two. Set the cross hair in the mid-point of the discrepancy by turning off this I· half distance on tbe micrometer screw. Now adjust the . thrust collars U!ltil tbe mark cor.Jes directly under the cross hair and lock them. Now if the crank is turned back the same r:~ber of revolutions as it v:as turned forw.a rd . .the mark on the t2ble T should return to its pos• ition exactly under the cross bair v:1hen the crank is locked. If it does not, this is either because the first discrepancy was too large or because the adjust □ent was made toq~ear the dead center of -the table. Thi$ second discrepancy should then be halved. as before end whe!1 the table is again reciprocated the ~a~k should co6e exactly under · the cross hair when the crank is locked. I The dead center for tte table then is bal:f the nunber o:f revolutions of the crank back fron its present position. It is evident now why an even nw!i'ber of revolutions should be •made. The n11.;."!lber could not be halved othervdse. In order to find the position 90° away from the dead center merely rotate the cranz 18 times. 'Ihe adjustment of the thrust collars on the sha:f't X controll the parallelism of the rod D with tte guides EB viben the carriage is in t h e initial position. This as explained atove is necessary to facilitate setting the fivot E in tbe proper place for the scale of the riven curve. ~he pivot Emus~ be a~justed to the proper ------ I ' l i I i ·31 place a.long its .guides so as to bring the point 2 ,;'; on the curve under the tracing point whe~ the carriage E occupies its final position. After ~uing this adjust- ment the point e = 0 of the curve would no longer be under the tracing point in the initial position of the carriage F were it not for the initial parallelism of these rods. Tbe following process is used to set the thrust collars so that this parallelism shall be obtained. Set the carriage Fin tbe initial position and lock the crank on rod K. Slide E as near the pivot under table A as possible. 1~a::rk a poi!lt on table A and. set the cross hairs of a micrometer microscope exactly over this :point~ Fov.' slid.e E as near F· as possi b.le. Unless the rod D was already parallel wit~ i~s g~id.es the table will move sliehtly carrying the mark away =rom the cross hair in the micrometer microscope. Now adjust the thrust collars until the mark is bro-._;_ght back under the cross hair. l II • Lock the thrust collars. A as possible. cross hair. Now slide E back as near This may □ ove the ~ark a~ay from tbe If so, orin5 tbe cross cairs to tr.e ~ark with the microneter screw. near to .? as possible. ?ush the pivot E again as If the r:ark e.gain !:loves ai, ay the •f irom the cross 'hairs aa. ju.st -':::<:s:: t'hrt;.st colle.rs as befo:-e j until it comes back. Conti~ue this cycle of processes until the rr.2.:rk cloes :r:.ot move appreciably under tte cross j I ·1 -- ---J, -- - - - - - - - - - - - - - - - - - - 32 two or three rep0titions will ever be necessary. Reme~ber always to bring the mark 2.nd cross hairs together • 'l _by adjusting the ~icroceter microscope when Eis near A J and by adjusting tbe thrust collars when Eis near F. l i If this rule is not observed the discrepancy will tend to increase instead of diminish. If the thrust collars are firmly locked with their set screws there is no reason "by the machine should not run indefinitely without needing adjustment. The wheel of the planimeter has two hundred graduations, each hu...~dred of which is numbered by tens from one to ten in the same directions around the "heel. The whole revolution cou.,,'1ter is turned b:->,~ a ten tooth pinion • meshing a sir.gle thYead ~orm and has twenty graduations upon it numbered. from one to twent~ 1 • Thus each grad. ua tion represents a half turn of the plani~eter .,heel, that is it represents the :passage o:t· one bu::1d.red wbeel d.i visions past the vernier. In taking a reaa..ing then it is ffierely necessary to take the figures off the see.le o:f the whole-revolution-counter and the remaini:rie three fi 6 ures off the plani:11eter wheel. In t~king tbese re~a..ings it is better to start with the wtole-revolutioncounter so~ewhere near ten for the initial reading. because the fir.al ::ceading is just as likely to be less the.n the initial one as it is to be greater. This depends on whether the amplituae of the ~er=o~ic for which tte .i j l ; _ _ _ _,-,,,d - I I 33 !I J f I curve is being analyzed is positive or negative. If tbe i initial reeding were taken near zero the counter might I. rotate ba.cJr.:a:rcl through zero with out being noticed by the I opera.tor and then ,'.'hat would appear to be a very la:rge positive reading would in fact be a small negative reading. The tracer a.rm of the plani~eter can be varied slightly in length as can be~ seen in the detail drawing of the planimeter. by ~oving the s~all block containing the pivot hole to the desired position. This cali- bration is best naie by causing tbe pla.nimeter to measure the area of a. circle of one inch radius. . .,_ ":'ihen J. I., is pro- perly adjusted. the pla.ni~eter will read one for . this area which v,e kno7, in real: ty to be 'G • A sir:-;ple method o:f causing the planirneter to measure such an area without removing it from the rod X is as fallows: for root mean s<2,uare ~a.lues. :lake a set up Place the pointEr Z exactly over tbe zero line of the table A. ~ithout moving the pointer Z rur,. tr.e analyzer through one conplete analysis. If the reading on the planimeter aoes not change it indicetcs th~t the point Pis exactly over the center of tbe disc Q v;hen tr.e pointe:::- Z is on zero. .I I reading changes it may be that t h e block Bon the Rod X has been moved or that the pivots of the V-erooved pulleys carrying the rod X have been chanr, eQ so that the rod Xis no ii longer over the centG~ of the disc Q. l, should be corrcctec. before atterr:pting- to adju.st the pl.:an- \ imeter itself. l I i 'j 1 - - -:- 1 • In eit~er case this ~hen t h e plani~eter finally remains _ _ i. -,l . l I i i i t r stationary under th~ above named conditions set the pointer Z exactly one inch away from the zero line on table A and in this :position ma.ke or..e comp lete R.:i.S . ana.l:rsis. The :pla.nimeter should register a difterence c;-,:.:~: of :fo-u~ s~uare incb~f· If it registers high the arm of the plani~cter should be lengthened slightly. If it registers? low it should be shortened slightly. For The Analysis Of Curves >r- Any curve to be ezpressed completel~· ( the..t is f:::,om - oO to + oO ) by a Fourier's series must have a continu~~ of periodically recurring values. I f I \ I l l By this is meant that if the proper length were chosen with a pair of di vide:cs, the poir..ts of these cli vic.ers placed an:,:-whe::-e elong the 9 axis of the c~rve would locate ordinates identically equal. If a · curve 'Vle will caJ.l this length a period. a. oe s ,not ful:fi 11 this condi ti en it is useless to try to f i t a Fou:cie:::-'s series to it throughout its entiretl. !:ny finite a!'.C0'.1...YJ.t of such a ci..L"'"'Ve c:::.n be e:s:- pressed as a Fourier's series hov;e-ver. '.rhe fir_st step then in a ~ourier analysis is to decide exactly hhat portion of the r,iven c"..rrve we d.esire to reproduce. If ~'he curve is periodic obviously the reproduction will be most effoc~ive if e p ortion of the curve one period in le~gth be chosen. ~e will c~~t a ny consideration of non-period- ic c~rves. lt is a.s 35 of a. periodic cur~e ~s one □ igbt sufpose. Sometimes ~hat appears to be a period of the curve is merely the period. of some very predominant barmo~ic. Careful examination •will show that ordinates sepa:rated what ·. one imagines to be s. period. are not exactly ec:ual but are perhaps . dePlate VI shows an excellent example of this. only If a small po:rtionr1 of this curve were seen, the a.istance creasing. .M3 would im!Dediately havr been assumed. ( and v.:rongly) as l the period.. In fact the.period. is as long as CA since it is only u..~til this distance r.as been passea over that 1 the va. lues of t~~e curi::e begin to rq)aet. This cu:rve as illu.st:n~.tea. 'has no fund.e!!le:ntal nevertheless it bas a neriod wnich must be made the limits of reproduction if a reproduction to infinity in both directions is desired. Having dete:rr:iir.ecl the :period of tbe given curve i t is 1 next necessary to decide ~bat p~rticularfpoint on the J I curve v. ill be called O. and. ,:hat point 2 ~ • • v!tat v;e choose tbere is a Fourier's series which ,;-.-ill re- 1 \j present it. No ~atter • s:'r..E: question is, n\Th.at point •will give, i:f used.,es ze:ro, tte sim:plE:st resulting :B'ou.rier's se2·ies? 11 P.Al:'SIOt OF' A CURVE I? T.SAT CUP.VZ IS SY:·,TIS8T?.I CAL .A30UT , TE3 ?on:T ~ ./lS P.. ?OlE. Obviousl~ it is desiracle to choose the initial point of the carve to 'be 8.:C::cl:::ze•:: so "cha-: i::f possible tte above of the ) _ _ _ . - -.... ~ - - -- • . • ·· - · - - - - - ·· - .J>- - - - -- - - - - - - · · - ~ \ ~: ~ .,....Q) \l ....... k_ ~ ' c:_, ! 35 j cu:rve should be chosen, if possible, so :.that every ora.- I, .. . inate on the left of the center or 1eo 0 point will be equ.2.l and opposite in sign to its correspona.ing O!"dinate the sa.rne distance to the right of the center. Hot all cu.IT es can be made to fulfill t'hi s but nee::rly .! J I I ree;,uire □ ent all alternator curves can be made to do so. This property immechately eliminates the neces_sity of analy_z ing for cosine tenis thus cutting tbe wo::ck exactly in half. The following ia a proof of the above proposition . If the cu.:::-ve is symr.ietricaJ. about t1:e point,, as e.. pole this means that ½ f (9) = f(2s-e). - Az sin 2(2~ - e) cos 28 ... ,,.. * "" - Bo - B1 cos (2-;:-:- - 8) - B2 cos 2(2 - -- - 8) - Simplifying the right tand member, B0 . - Bl cos 8 B2 cos 28 Since the two members are ide:!:tica.lly eq_ual the terms of li~e frequency are separately e~ual to each other. the only value ,:nicr. the coefficients B 0 , B But 3 ._. , et C. can have to ~ake tbe separate cosine ter~s on either . side equal is evidently zero therefore there are no cosine terms. Ea.vine thus located. the period poicts o:f the cur-ve to be analyzed it is desi rabl e to know w~ether there are even barrnonics p r esent. ----- • ) ·--- - - ·· ·---- - -- - - - -- •••• 1he ortinary altsrn~tor ra= e!y - --·- --· -- -··--------- 37 perhaps never, gives even harmonics. In order to ascer- tain the absence of even ba:rmonic s . slide. the curve 'Ti I"\ untis to the left until it comes directly und.er the left· half. ..,.r.e If tbe two halves e.re ther. syr;Eetriacl about -4-" e e.xis the curve cont~ins no even bar~onics. proven as follows. or The above cond.i tion is equivalent to · f (9) = stating that This is - f (e A1 sine· A2 sin 2e + sin 2 ( e + 'ii ) B2 cos 28+ "' +- "' * * - Bo - :s 1 cos ( e -:-- 11 ) - :s 2 cos 2 ( e ..,. '7 ) "'" * * * I Simplifyipg the rieht r.a.nd me~ber, A1 sine~ A2 sin 28 ~ sin 28 ~ ~ * * A2 - E2 cos 28 ~ * ~ * Since the expressions are identically equal te:rms of the same frequency are separately equal. j :But tbe only values of A 2 , A__,_, E:tc. B: , B:,, B. ~• etc., that can !:}ake the even ' --' nu~bered terms in the right hand member equal to the corr- 1 esplnding terms in the left hand me~ber are zero. There- j fore aJl tee even terns including tbe constant term j ,-- I i l l; are zero. These tvo ~ests canoe eesily made by inspection &nd are independent of each other. If they are both fulfilled the knowledge whi.;b the::/ give se.ves three-0_uE1.teres o::t' "'cbe v1ork which o"'vherz,·ise would. be necesse.ry in the an8..lysis. j Testir.:3: the ho..lf cu:rve :for. s:y"r::mctry about en ord.inate - - - -·-·- - - - - . { Z8 ,c-- set up at the point - is sor.}etimes suggested. It is not 2 however as eeneral as tbe tests above given. The fulfillment of this test indicates the absence of even harmonics ;provided there ere no cosine te:ITJs. If there are cosir..e terms it indicates absence of even sign harmonics and odd cosine harmonics. Occasion might arise when this test v:ould be val'.;.e.ble. Tbe curve might fulfill none of tbe tests except this particular one it which case one half the wo::::-k of anal:ysis ·would be a voicled. Such could harc.ly I be the case 'however "i th curves resulting f:ro!:l alterneting I current generators. . !':ROC:SEJ)URE ... . IN S:S'.i:'.rING UP l j ~ end 2 ";;' on ,. the curve . to be analyzed. cut o~f a length of the oscill- j ogrs.m about 7ft inc'hes long to incluc1e tr.e zero. Raving decided. t}le location o:f ze:ro. 2 -~ poiLts. I f and The zero point should be about three-quarters of . fn incb from one end. Lock the sha::t K b.y pir..ning its cra~k to the cast iron bracket. Place the carriage Fin j , its initial position by sliding to tee mc.rks on tte guide l rods an·a. dro p ping tbe ~alf ~u.t into :place on the sc:::·e.., . !/ 11 I l i i -;7 I:To,1 take t'he section cut frc,m t'he oscillogram cJ.nd. secure it to the table .A ,r:ith th'Q.!'.:31:, tacks is sucha position that the e axis or "zero lin e" cor1·espo:nds with the guicle ~arks on the s ~all br2ss inserts at either end of the table. Stc.n'.liY:E; 8.t t t e t e.'b J. 0 ~\ CY: ::. of the c:.nal:yze:r facin g tl:e machine t?-: e p ositiv e cl:.. :re c t :.oJ -on the e axis o:f tr.:e cur,re r i - -·! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~ l i 39 j lI will be to tbe right snd tbe positive direction of the I I half nut out .of the thread end slide the carriage F to I ~rods. .1 ordinates will be away from the observer. Next lift the its extre~e position according to the marks on the guide Drcp the half nut back in the threads. Theos- cillcgrapli curve will have moved along its. guides but the 2,... point 'tvill probably not be under tbe pointer z. Loosen _, ' ___/ the __thu.111b--·sc:r-ev.' U a:1c slide tbe carriage E until the 2'Tf' point on the cu:rve comes exactly unde:r- the :pointer ano. fiz tbe pivot in this position by tigbteneing tbe thmnb i screw. ?.eturn the carriage F to its origir:.al position. _.. J I NoY, lock the shaft L by means of its crank. havir:g first turned the disc to the position marked sine or cosine, dependine: on the term d.esired., by meens of the crc..nk. 1:ext set U!) a gear ratio between the shafts E e.r:i L er;_ual to tr.e m.L'T.:be:r of the harmonic desired. 7-be obtainable with the gears fu:rnished with the analyzer: :Ratio. 1 2 , '7, V -4 , f 5 6 7 1 Idlers. K 20 20 20 15 16 20 60 60 -:.2 0 ] ' .8 l 9 20 -, .l v • i 10 12 14 l5 16 l j ' 1 'l l i I ! - ~-- ~::. 30 : .~o 40 / 20 -, "' ..Lv' "' .,.l.J 4.0 40 30 20 20 60 60 30 20 60 16 20 20 15 ~ 60 "1 r:: 60 60 60 J_ V 16 ,.:.v"' •• +- i~ O.., E: 0~ l_.., i:::. ~o. ,. 4 0 ,: ,:-,·· · '.-.; 60 ·, 60 60 60 70 60 60 40 60 70 60 60 { ------- - { - - - .· : :' j ·l 40 l j l I To obtain the eleventh an& thirteenth harmonics l- v.,i th tbe gears su.ppJ.iec. special ~etting of tr..e carriage f Fis reQ~ired. l Instead of sttting carriage Fon final ~oint as per instructions for adjusting E, it should be set 2t point ma:::-kec. 11 if the ele·venth harnonic is . de- si:red or en 13 if the thirt eentc harmonic is desired. In rraking the an~lysis it should be allowed to run no l l 1 I I . farther than the ~ark at ,..-bich t'te settine r,as made. :For 11 e.:r:.d 13 the sa111e gear combination is used as for 12, that is 15, 60, 20, 60. This could be avoided 1.• .t·..L special gears were cut to give ratios of 11 a..~d 13. har~onic can be obtair.ed by useing special gears. 1 . The c::::-an1-:s re~ain locked v1hile the gears are being j secured by means of the thumb nuts the.t hold. them by l friction. ,j ¾ J t J J:-.y · ·\sei that the planjmeter reads about nine or ten on the counter disc. Take en exact readi:q; o:f the pla.nirreter with the pointer Z exactly over the zero point of the curve. Kow unlcc1: the brass c:ranks e.na.. start the e.ne.lyzer either by turninf the crank on shaft L, or by startin5 the ITOtor on tnat sese shaft. As the table A moves keep - the pointer Z exactly over the curve under test until the I l l i .L· • 12..-rr-' po1nv • ,._.1s reac '..h ' eQ, then st op irrm::ea.ie. te l;r. 11 ..LJ..na. If the ordinate o:f the curve is not zero bere :eove the pointer l, i l !j !l -~ · planimeter er.cl tc:ke 'c'he di:'j·ere:::'1ce between ..,__, ._,r._e i'irst read.- ----~l .:,- ,, - - - - - - - - , - - - - - - - - - - - ' - - ----- l] ·i - - -- - -- -· 41 ing ~i-~ . this :fjne.l reeding. If the difference is neg- ative the result obtained shoTis that there is a negative amplitude of harmonic of the order just anelyzed for. positive, the amplitude is positivE. i 1 j j ! j i If The reading of the planir::ieter is the amplitude in inches multiplied by tee • r.umber of the harmonic sougbt. ' In very accurate work the process may be repeated as often as tesired, the carriage . ... . ? being returned to its initial position without vurning t'he she.its. The initial and final readings can then be tal::en. ancl their difference di vioed. by the number of the I 1 l harmotic and the number of repetitions. 6..e&·ree of accuracy ce.n be o'bta.ined. Thus elmost any Tbis device is hardly necessary except when tree e.r.1:pli tua.e is small. j l t ~ i' ·j., . E.f:...c"q:-tmnc SOU(}ET GIVES T?E J:2.C?LITT.JDE OF TEE F..AP'2\WlHC I:iJ IECEES. 2aving determined the amplitude of as many harmonics f j as isthoufht desirable it is in order to deteITJine the 1 ::Cear: Squ.aYe" val'J.e o:f tr..e curve in ar.1peres or volts is Il known fror.:i tte rce.a.ing o:f an 22!neter or voltmeter taken l l scale of the curve in ampeYes or -volts per inc'h. at t'b e time wh en the o scilJ.ograrn -v,,as made. The "P.oot ~he ~e:xt step is to deter;nine the ::~oot I: ean Square" value of t::.e curve in i::iches. l ____ ___ Tl .l . ' . f f ! l J 42 ?.ernove the long rod Z •and vii th it the ulanimeter. \ The pla.nirneter may be li:fted. f~om the pivot on the outrigger of table T ana. foldE'd. und.er the rod X v1here it can be secured fro!n swineing. 1 The rod. X is tbe::n removed frc2;1 the pulleys by swingine it up end. over out of the wa"j-. ~emove the taThle T and mth it the small steel pins l Tihich will slide out of the bole in the bronze disc Q. 1 l i l! 'l .1 l! f I Secure the irre 0 ular alu~inuro casting upon the top of the disc ·Q. ~he cxcct horizontal angle at wcich it stanis makes no difference . Replace the rod Z on its pulleys. Unfold the planimeter aria attach its pole end to the pin in.the outrigger of the irregular alu:r.inun casting · allowinf the we.eel to rest on the projecting table of same. Now start the . analyzer with a one to or.e :re.tio on the gears as though an analysis for •1 fundament2.l 2..mpli tude were being mac.e. 1 with the pointer Z as before taking initial and final readings on the planirneter. I :Follow the curve The difference of these read- ines is t~e sean squnre ordinate of the eu~vc i~ square i inches if the letter ~us been traced over but once. f root mccn square or &inate is evidently the square root of tr:is difference. The 'Ibe root !!€ml squ.e.re ordinate in nmpcres (e::-irr. eter reD.cl in g ) c.ivi cleci b:? tr.e root mcnn sc:u~rE' ordint!te in i~ches ~ivcs t~c a ~p e rc c ~er ~nch scale of tbc curve. --____j --·-· - -- -··-· l - - - - - ~--- - - ·-·- - - -····-·- ··- - --·-·- - ····- - - - - - - - - - ~,. 43 j ·1 I ] j l ,1 ll ·l I Then all of the harmonic amplitudes v1hich have been found. in inches can be multiplied by this factor to get their /1 rn pe;re-s value in ~ . As nearly all formulae applying to al- terneting sine waves of current are constructec for effective or root mean s~uare values. it will be the~ conven- ·./ · , ient to change the values of tne amplitudes (or maximum values) just obtainea to effective values by dividing tbem 1 l lI '! by 1.414 or 7/2 . It is rarely of any importance to know the relative phase of the barmonics of different frequency in power Tiork.It is t!'leir size in 'i'7hicb. one is mo're often interested.. If both sine and cosine terms of the same order or freQuency tave been found by analysis their root mean s~uere or effective v~lues may be combined by taking the square effective root of the sum ?f the squares of the two separate values. This result is then tbe total effective value of the partic- ,. ular harmonic in auestion and may be used in impedance formulae, etc. : : TlrACER ARM ~l • /JliaH t: on~. ---:;m; ..... I 3 6-'r • ..5 fuel. Make one I • I ----- - -1+I i-----------6-----------4-1 ~ .32. ...L ~ /6 t--+--+--------'11 I i ½!tXJ I t ?irc!J :=. 06 8 7 Vernier Make one. 2 00 Graduations. on ~ilver. ff0LL !NG WHEEL '0'30/< . \O~~\l/vu>~ -...:: // 11)- ..___,......_........,,1 if. fiole __/ i 'l~ -u, • ~II\~ - lJfT/1/L.J or ?LAIVI/YlfTflt -;J//fo[)f' [tJiL£6f (}/" 1£?/IIIIJLOGr • ' Pa5odener . ~ I O 6~ If ECOfrD/N{J ])!5C -- - - - - - - - -- - ···-· - . 0 /J1oke ol7e. J,1dre one . . .. - - - -- ----- ------- -- -- - - - ---- - - -