Deforming of the Front of Travelling Waves Along Power Lines Under Corona Conditions Citation Genachte, Paul Frank (1935) Deforming of the Front of Travelling Waves Along Power Lines Under Corona Conditions. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/enzk-p071. https://resolver.caltech.edu/CaltechTHESIS:10112024-222723327 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): Lindvall, Frederick C. (advisor) Mackeown, Samuel Stuart (advisor) Thesis Committee: Unknown, Unknown Defense Date: 1 January 1935 Record Number: CaltechTHESIS:10112024-222723327 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:10112024-222723327 DOI: 10.7907/enzk-p071 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16790 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 17 Oct 2024 19:13 Last Modified: 17 Oct 2024 19:13 Thesis Files PDF - Final Version See Usage Policy. 29MB Repository Staff Only: item control page

DE:b,ORMlt,.TI ON OF THE FH.Ol?T OF TRAVEL!NG WAVES ALONG l?OWEH. LIMES UlrDER COROUA COJ::!DITIOlfS THESIS BY : PAUL F • GEllA.CHTE IN :PARTIAL FULFIThIEllT OF THE REQUIHJTir;::IDrTS FOR TH.El DEGH.EE OF DOCTOR OF PHILOSOPHY CALIFOIDIIA I :tISTITUTE OF TECI:INOLOGY , J?.A.SA.DEllA• CALIFOIUJL/i. 19}5 The author feels that this thesis would not be complete without aeknowledging his indebtedness to Professor Lindvall and Professor rEaokeown for their invaluable encouragement and advice, and gladly takes this opportunity to do so. -1- Deformation of the front of traveling ·waves a.long power lines under corona conditions. Introduotion. Many experiments have been performed on aetual and artificial lines in order to determine the attenuation of traveling waves, especially at their front. Our ob3eot is to try to derive an expresaion,. depending on the pa1·wneters of the line, which will give us the inforraation we desire ooneerning the shape of the front of the wave at any point and any time. From the knowledge of the maximum gradient of potential, the engineers set up the pi•oteoti ve devices for the apparatus oonneated to the end of the line. The telegraphists' equations, modified as we shall see, will lead us to the result. As a. method of handling these eq:u.e.tions, we have the ohoiee between . the classieal :method and the Heaviside operational calculus. Ror networks with lurnped parwneters. the classioal method of treatment .o:f transients is easier and does not require the symbolie operational caleulus. For networks with. dis·tributed pa.re.meters, hQwever, it is alm.ost impossible to reach a. solution with the ela.asioal method .. This is due to the fa.ot that the voltage and current distributions depend on the boundary conditions. The :physical truth is thus disregarded since the waves, traveling at a finite velocity, ignore the terminal oonditions until they reach the end of the line. The operational oa.loulus considers the waves on their trip a.long the line, and is therefore the natural me·thod to use, It cloes not ooour now to ele~trioal engineers to use anything but the vectorial or imaginary symbolic method to solve steady a.o. problems in all oases. In a similar way, the operational oalou.lus will become more and more the symbolic method of solution of' transients, ancl, in f'a.ot, it is the only natural method of investigation of transients along lines. Division of the work. We are led to divide this work into two parts. In the first part, we shall lay dovm. a sound basis for the Ol)erational oa.loulus, This will require the thorough discus i:, ion of the Fourier integral f.ormu1a.'hion, on whieh a.11 the operat1ona1 oaloulu.s 1s built. ~he seoond part will be oonoer.ned ,vi th an applioatton ef operational calculus, i.e. the deforint.t."ti.on of the front of traveling waves under co1·ona cond.1.ti.ons. First l'art. The Fourier integral formulation. Let f( t) be a s ingle-valued £\motion having a. finite number of disoontinuities and of rn.axi ma and minim.a between , +4 , and m,.ch that exists" / . ~ ~ J j di ,() Then the function :f(t) can be expanded in a convergent Four ier s eries: wher :e :. w Thus. for all values oft, we have: whioh represen·ts ·the harmonic analysis of f( t). Transfo rmation of the Fourier series. We know t ha ti Let: n in·to (-n). For /J1. =": with: These two relations (2) and (3) are most important. They are the first form of Fourier transforms for -.5funct i ons having a. f:tnit~ 12eriocl;. If we draw~Jva;·w , we obtain the disoontinuous fre({½enoy SJ?e<rGl"'ti..'111 of' ·the :t"'\mo-tion f( t) w1. th a fi.ni te period • This 11 frequency spectrum " gi vea the relative importance of' the successive harmonios 11 c.. (r..,) , ]\motion with infinite period, i.e. having no period whatever. Thia oaae is the limiting one of t.he former rela:t;ions. when s tends toward i:nfini 'l;y, We had, for the period,: When s tends toward C)J; . , 1£ tends toward ~ w Let f denote the frequency, then: and : a.nd.todoes ~ -6In order not to confuse this :new t\mct:i.on f( t) with the one having a finite :period, let us denote G(t) a f\mction 11a::::= ;mu:7-~~;;'.~/Jz~-tY~ -oo with F?t.trtf) = _/+~(A) E ;;J·.ty7j-A -oo The rela-tion (4) gives the harn.1 on1o analysis of the f'unotion G( t), whereas ·the relation ( .5) is the continuous frequenoy spectrum of the i-unetion G{ t). 7'{/~f) gives the relative importance of any term of frequency f in the harmonic ruialysis of the given func- tion. In other words, the equation (4) is the a..~alysia of G(t), and the equation (5) is the synthesis to regain G(t). ro ,«ti) The J:"ela tions ( 4) and ( .5) •are Fourier trc1.nsforms and will serve to justify all the steps taken in the opera·tional oaloulu.s ~ We recall that the ~ela;t:i.on ( 4) is limi t ed in its a:pplioa tion to a :f'Unotion G( t) suoh that :/+°/i;-~.(dt exists. -ex, This implies that G(t} goes to zero when t beeomes large, -7Change of variables,. Let then (?) vnth :pis a }!llre imaginary quantity and G(t) is altogethel" dete~~ined by the behavior of F(p) along the axis of imaginaries, (.9) Jdf- The expansion ( e ) converges only i£ / ~ o exia·ta • If' Wfit anal~rze f\u:•ther the relation (?) • we can draw some oonolu.sions a.at<:> the nature of the fl:i.netion F( :p) by examining yJhat happens vvhe:1.1 the complex p is equal to infinity, Let pin the relation (9) be equal to a oomplex qua:a.ti ty : ~ 0 Then, our relation ( 9 ) becomes: l'/JJ =/6'(,1) ~-\"'~~ -.>.!',/~ dA 0 Or oour se, £ C9) ciA/~"" is always less than unity• FfJ ~ /"'vrv z-'Y'""'dA {lo) 0 ~he lntegration in ( 10 ) is :perf'ormecl along the real axif: o:t thel - :plane, and~ is :positi,re. I f / goes to~ • the integral ( ~O) goes to zero for oos°'>o, :t.e. for points :P a:t infinity in the regions I an.cl IV, or f'or x. :positive. In order to f'ind out what happens when oos rx < o or -l: < 0 , let ua replace ~ by - ~ , then: F(/') = -/-C(.. ').) z: ),,!"'"""',: ¾l~d ,'.I {J ") C> But G(A) is zero fo r negative v~lues of the ar&111ment and £ .\ ;., '4rl (X 1 is zero £01.. / ' == co I and E ~ o,( ~ ,,i_ ~ ~ .., /'- -;r- 1.eo for -X <o. Thu,a, for all J?oin'lis at 1.nrini.ty : ?;;.,~ =~ = o Going baok to our r elation ( B}, ilt order to investi ... ;,rga ta how F( p), z. 11ehavea a t all :poin ta at :I.n£i ni ty, we write: J1(;,J , ~ P~ .l'(j.J). z x~ £. cl; z: zero if x-t <o and this will be , i,e, for ~ >o with for ,x < o vii -th and t <o i >o We arrive thus at this impoi..ta.nt oonolusion: ·the in·tegral ( 8 ) along the axi.s of imaginaries in the p-pla.ne oan be renl~eed by a oontou.r integral composed of the axis of imaginaries and a half circle of infinite radius to the left for '->o , and a half circle of in:f'ini te radius to the right £or t,o. 0 Our e Q:'~'ta tions ( 8 ) .:md ( 5) ) become now: q/t) = ....!....,_ L 7'/f,) z ;0Co/ {ii) Tff') =- /oo(r(,l) ~ -,P~.£.:\ (i~) .tlt/ J 0 with ti/2) ~ o/ t < o and the condition trmt /""81&'J}~ 0 exists. we are able now -to evalua.-'Ge the contour integral (11) with the help of the thcol""IJ of residues. If the function F(p) has a pole at the origin. i.e. I ~ 6 ./h 1:rF/41:t. C(A)d,'.\ =<X> . • aineeJ._ Pf),)£ o/" 00 ( o - must CA.l/CO be zero ., we have to e.void the ori gin by a small half circle around 1 t in the right he.1:f' plane. This, of course, corres- ponds to writing: Theory .of re,s idues. Evaluation of ~he residue a. t a : pole of an_y: ort,e,r .. ;J.. --- p•pfa..u 0 In Ol"'<ler to use consistent notations, let ·the complex plane b·e sp()ken of as the p-plane. Let 'f//J) be a complex function : f'~J == Ffx ~cf)= ;J/(x.✓;) +-cf :rf~;J but Jf' Jj" )Jr ' °'2>.I ~x - ~I' = ~x + ~~ ~ _ ~ , _ Jfi + , )z cJ v- - -;I - ~ J J/ i)J Assume that fj?J) ia ei~le.. valued and continuous in a eloaed region R. Then, all _the derivatives ot..R~J)a:.fi.d :M'J') wt th respect -'Go x and. y have a. meaning, and we conclude at onoe :. )JF ';:,I j~ = ~ 04-) Thus: f'~J.f = (.fl 7;;1·IYdx 'l1J) and 1;-) r{,6)~ =_fr$dc-Ij-) vfo,/.,; +A~) .f == (.B. d.t: -.z-j) ~(z.L.r. +Jr But, beoa.uae o:t the relations {14) • both expr&ssions under the integral sign ill the right hand member are exact differentials, thuli I f '('(/') dj, :- o It appears a t onee that: The value of a line inte. g1..al about a closed path in a given direot:i.on is equal to the sum of the integrals a.bout small clos·e d paths in the same cl!reotion. each enclo$1ng one point of diseontinui ty i:n the original area, Our next step is the evaluation o:t' x·es1duea, 1 o Lemma s Cauchy' s Theorem. Let f/lJ) be an an~lytie fu:notion ( continuous and sing1e ...valued} inside of c. Suppose f'j/J) jJ-a... analytic bc,tweEtn C and c, o being a. small oire1e Of radius r a.round the point :p =a . :J. \8 c ;£_,.,,-/ X 0 2 . Theorem. Ta.ylor' s seJJ1 es. Suppose f (~ analy·ti~ Within a eirele of r$.diua R. 0 We know that, then: This is Taylor 's series. It is q:u.it e gen~ral, aa it holds fo r complex fu.netions, l:f' a funot i on nan be expanded abGut a poin t a, it is ea.id to b$ regul.a.i.. a.t th@:ti point. 3, Polee and resiaues. ~ analytic f'uno.tion f'/1/-'J is said .,GO have a pole of order m. at a. point p =a, when: Thus 'f()J is a regular function a.t 1> = a, and ean -14- therefore be expanded in a Tay)or•s series, as: ,L 'I'I?J ~ fft,.) + ft A_) r 'I) + (j,-t! ...) t fa} + A Consequently: - - - - ;r ·+ efff'//') o/J The integral along any eonttl\Ll' C enolosing the pole p = a, is equal to the in·tegralf 'fftJo/ along a small oirole of radius r around that pole., ·& Then : /' - djJ 1? and ----:- {;1 so 4:: -y:·· ~ =/ (/'-4).,_ = -2. £: /•e = ~ Lcl~i d6 4- . 11. /flA,u)GJ 1-11e. d6 Z .:-, . [ ~ =- - - I - #1. c/·(t-ttc.)~ - ' J For m::;=1, this integral is zero since r : o and / '(i - ,n.) .UC == I L For m: 1 ~/'~£. = J/·cf,9 = o!~/ " (: On the other hand, the terms groupect under the s.ign II will oont;ribute nothing to the integre,1, since these ;:) . o/ terms fo:r:m an analytic expreseion, thus: f C f'/J) (,fl(-/) = !' ( /jH. . ca.J -t) .1 ..t l{_L'. {7 (,Ht.-'.) 1· I" (~) ~he qu.eJ.lt ty (~-()/ · .e.s tht:i • '1 i "' which en ·y ooeffie eXh m in the expansion ( 16) wh:loh eon.tributes to f C FI/J) 3/J the residue of the po1e, let us denote 1t a. fl (-#l-1) {4) but (m-1),I {IJ) of ~ order \V-l thin C, then: 0 G) the sum of the residues. The expression (17) is very i mportant: it gives a direot and always simple wa;y to obtain the residue at a po1:e of any ord.er. For a first order pole: m : 1 , thus, _;f = -l'u.;; = J ~ [ r; - a..J FrJJ> which is well sinoe o:. =1 knO\ffl relation. We oa.n now go be.ck te ()Ul" reoiprooal relations (11) and (12) and :proceed £urther. As these eqlle.tions stand, they give either G( t ·) o~ F( p) When the othe:t :tu.notion is gbten. If th~ complex ftu>.o- tion F(p) is given.. G(t) is Obtained by a conthr integ:..-a.1. Sinoe G( t) is ze1.--o when t is negative, we eon:clude that the :funotion F0) . E ~r ha.$ ~o poles ;n the :n~t h~lt :plane., When t is poai tt ve, we he:ve : Con sequently; G( t) is eq\W.1 ·to the B'W11 of the reet- dttes at the dif fe:r:ent '.&oles of ·the f'u.netio:1.1 IFu,). z ,/.'1· We have, in th0 e;q,ressio-.a. ( 17) • the means ,of el'ft,i• luating th0se r esidue.SJ at :poles of W:rV -high order. The :\. tfilttegra.l ( 12 ) oan b~ reduced to two rea:t integrals. Thus; from t he knowledge o:f' either F( p) or G( to), th~ otn,e;- i~~tion •¥ b~ . ;1.7~~a.;t.l obt~~n,~d. Bal·th. V&.:n De:i:- Pol, 1n a pu.'b1ioation in the l?hilo... actphioal ]JLage.eine, gave ln 1929 a. :really strong me.theme.... tieal basis to ·ohe Ql;1!i.81ne.1 eonoept1on of Eeaviside11 The entil'"e theoru of the op0rational o@.leul.us t .s based -u,pQn a few rem.a.vkab1e p;coperties of the int;e g;ral equertions ( 11 ) a.nc! { 12 ). In ordtr to d1socver these pro:pert~es, it is neoes-.. ae.ru to make a eh~e of tu;p,c.,i;iion, reple.o1ng F{ p) by !:!f!!.J • ~ (1.9) In a aho:rthan.d. ~ashion, we ·write: whioh we read : It( p) is the symbolie representation or is ,.r::tte. ,.· : . · •11 ·... ;!,O T , " t ·, . ·c _ t . P Sl!;bG,l,i~a~l~ eg.1,~Jve.l!!}t to G(t). We may also say, together with Van Der J?ol, t .h at H(p) is the i~it, of the o~-ii~l .. 1 · ' . ... ,• . . • fun. ot:ton G( t). ~he expression ( 21) is called an oi,era.tional ~ . . . ---..fl(!,) or F( n) is known as the LaJla.ee transform o;t G( t) . e~tioll., ' ~ 8 ' ' ., . . , Sinee the equ&td.ons• ( 19) and ( 20) are rfu,iprooal, we shall establb.1h a f ·ew important rales on the basis of the equation (20 ) J .J{f;,J = jJ J /co '7/A) J -/';\ /, etA 0 1. Let. ~ =-~ • ! be1iag a po$1tive constant • .Ji/J') -: o/" /""'Gf,.t) .t" --)i ...:t,.f D thUa: Jlif.) ~ I /"'u/a.tJ i -,,.tdtc Ccms•qu.en:bly: fl(!-) ~ G(a.t) ~...Z) integrated by parts, is equal to: I [i: -rA c;. 0) + I' /c;. M z-;,~d.:1 {" Vfe have shewn tha. t G( t) oan l:µ) t be infinite at t ~ thus: or =o:> , The first term has the term a, c,C> • Using l' Ilc;,spi tal t s ru.le: -1'vm.[L1;r«Jdotl .t' ;,A . =- {i-foc,) A= CC> /'. X 00 and, as we lmow, <r~) must be fi3:1i te. Thus, under all oiraumsta.nces, the first term cancels and: or 4. In the same way, we e0uld establish other relations, euch as t I' /' , .Hfl' +a) ~ £ -a.~G/i:) (.._?~ +e1. which is the tfe~viside, £!hif,t1n~~ formula, Alsen ~-3/"H/i,J =f= [o c;.(i-a.) Conelusion. By looking at th~ operational equa.~ tione (23) and (24), we can draw importmt conelusions. Suppose first tha.t G(O )=- o, then :P is f.lqu.ivalent to taking the deri va;t ve ; f J dt-- . i .n tegral sigtJ. , whereas; corresponds to the Of oO'U.X's•·• i:f G( o) is not zero, then 0 there is an additional term in the derivative: t: The ae,oond te:t"m pG{ O ) 001."re~onding to the clisoon. C tinuity is called an impul.a e, In the same way: /'H(I) =:= :Jff +t[,-:::)L=: /tClo) ~ so , that:? re:pr-e sents ~~ ' di,,._ , :plus the i m~lse terms in the funotic:m and the (n-1) first deriva tives at the origin. _,,,_ represents .J../t-J t- - -- - / t....,,..,..,-. ~r / 0 D ~ ~ Conseg:u.ently, p v/htoh ie the independent va~iable 1n the image. beoomes &1. symbolic- operator in the o:t'iginal function. Thia property• which is the olu.e of the operational method, has given the name: Oiera.tional o~lcru.lus. Operational . equations. The fudam.enta1 pro~1em tn operationa1 ealcu.lua is to find the original f\ln.etion a:f' a given ima.ge 11 An operation.al equation is thus of' the type; C(i) ~.Hf/) where H(p) is a. B'iven function of p, a.lid G( t) an '\Ulknown function ef t. The solu.t:ton of an operational eqna.t1on redu0es i tse1:f' to the solution of the in.tegre.1 e~tion .. ( 20 ) , and CSt.."l be ~btained by the t:'.aeo:ey- of :residues. However, this solution o~ be obtained in most oases by reducing it to the solution of a certain number o:r fun- damental equation$ whi.oh we ,lhe.11 nov, consider. We have n-...· d. a1· rea.tty found the m.;an:tng of / LJ 4't ........ / h. ~. Fux1.da.menta.l e~tionfh obtained from : . 1. p • ·. ! •, 1 T .........-- . •tf:•IW'· . .JIii'J .,, /' /'c.r:i.> c _,.. ~d'A C, 1. Le-t G( t) be · equal to unity. Cf:) This i s the well- kno\m unit funot1on of t 0 \'Ie notioe at once that Heaviside. /.£C/l))"dt- does :not exist; 1. e. G( t) is not of s·ummable squ..ares. Su.eh a funotio:n G( t) is not representable by the Fourier integral as established :positive constant, is ot summable $qua.res, 1.e.the integral ... 22- Fourie~:- transforms . We notioe that tlle equ.a.tion ( 28) only . ......... . " ' ' . . has changed. The equation ( 28 ) merely means that,. when we inte&;rate along the a.xis of imaginaries , we must avoid the OI-"i J :ln by a. $'!}.all half eirele in the ltight half plane. Thus, if" G( t) =1 , the generalized eque..t ion ( 2 9 ) gives H('U.) • 1. The un1t f'u:notion is thus expressed by the contour integn.11 I"" .:;; ff~ e{6 and / .' / Cc.uieequ.ently, a. eonsta.nt is its own image. 2, Since we derive: I (Jo) lil:any mc:J:fe relations oan be found; i t is the objeet of textbooks on operational calculus to establish them. L.Ar-,»lieation of the operational ealau.lus to eleotr1e networks, So fal", we have built a logtaal ensemble of deduc .. tiona from the Fourier integral formulation. We have been led to the oparational ealoul~s. Now that the basis of t he operational calculus 1S finaly established, we pass from the purely mathematioa.1 point if view to the praotioa.1 applications of operational calcu.1tt$ to eleet.r -ioal eng1neering problems. The link between the f:o:rmer part and i t ·s applio(il. tion to 1?il0otric networks rests on the follo\dllg p:ro:peI•ty. Lst u.s consider the differential gtqu.e.t:ton of any high orcter vJi th oonstar1.t eoe:ff:1.ei ents: with the oondi tions: and where .E(i) =o for J=if-=:- - :f'or t=: o d"" drc;;,_ at f =o is not neoessai~ly zero, These oond1 tiona express that a. t t- = o , a disturbing force E(t) is suddenly ap:plied toe,. linear system _p reviously . in equilibrium. Lat us replace the equation ( 32) b y a.?J, integral equation. then the The i mage of the left han.d membel1' ia: The ima.ge of the right hand member ia: ;, /Z ~) -;>:f~ £ C I mages ( .:,., ) e.ntt ( 34) must be equal; -n ( a,,j' "- tf., f ,,,,,,,_, ;:..co \ ~ - - + '1-_.._y~ /A) E o -_;.,;\d~ :/co.;; -r~ I'cA) E c/ A C The integ1•al equation (35) takes the :place cf the di:ff oren tial equation (32 ) • Let ~ /4) ¥:A~) ,n. and 2(/') = ~f' _,_ "i /" ,n-1 + Then, the integJ;?al equ.at.ic,n (3.5) can be written as: -2~), £(;) =A/;,) ~ E ft) ?bus: 01• ;Jfl) -;- 1/$ -f(j>) = (.U J .N(i,) (.1~ or ;tfl) ~ ~ .l/1) Z(/) . Z(i'J We have also -:;JL ,· /3 A ;l:,J • and ao on. -at. Z {?) ~ p ✓f , ' ; -::;, 1 and BO ml, Consequently, we have reduced. -the solution of a. linea.x· d.i ff'erent:lal equation ( 32} \¼'i th equilib1•ium time boundary conditions, to the solution of an op6:re.tional. equation ( 36). This is the .ju.stif'ioation for ou developed. ma.the ... matioa.l baokground on oper•a.-'Gione.1 caleulum. Suppose E(t) = 1~ Since : and : I -:- I 1 rJf!J -:2 whioh means that: • M( ») = 1 U,,> .Z(;,) = / (sJ) j "" JI-~),£ -/'J ",;/). {.SS) Let us now oonaid.er what the symbo11• eUJJ,Ui.t:l.on (371 means, I t the f'w:ic.r~ion y( t ~ that we are trying ·co determi ne is a. ou.rren:t; • and i f the ill'.11 t function E{ t) is a suddenly ~-----a:ppliE:d unit voltage, then Z( p) plays the l..$le of an • i mped.anoe, Thus Z(p) is the 9!bolic or the operational impe·danoe of the oircu1 t, :f'.o r a suddenly a.:p:plied unit vol tage,~ This synibolie impedsi.noe is n.o t a clist-u:rbing new ·conoept; . 1 t is vei--y mu-oh like the complex impedance zy·u,) in the oaae of sinusoidal a,o. c:m.rrents. In the same ViJa,y, Z( p) gives the means to. ·ti..eat p:robJ.ems· in the non...steady state by means of equ.a.tions having the form of the Ohm's 1aw. The inverse of Z( p), . I tu.nct1on , and •• 1 20,) . , is ca lled the g~nerating /J~f/')• is called the Lapli!.oian. . . . . . tx-ansto~ . The current x·esporiae to a uni ·t voltage has been called the indioit1.l e.dmitte.?lee by Carson, and d~uo·ted A( tJ. thus: and: These e\tuatio:ns have the sarne limi. tations as before: 1 • A{a,) must be fini tet, which is pl:cy'sioally the oaae for dissipative netwo:tJka ( all the rootet of Z(p) a.re thus iu. the left half )lane• 1-v.i. ·th, even tu.ally J the exoepti on of the o:r-igin ) . 2._f[A/1)) di • a> 0 .J/ ' exists, The equation ( 40) ia kno1-,ri1 in eleetrleal engineering as ·the intini te integral theorem. It is the defini tlon. t .ha.t was -u.sed. by Bal th, Vf,U'l Der l?ol f'br ope1'"at Or$, Canon, in • hi$ 11 The9ry 0£ transient osoillations f>:r eleotrleal n.etworks" ( Trans. A,r.,:m.E,38 1 1919.p.34.5), ma~t:~s of this theo- rem tha oen tral :fea tu.re of opera. ti ona l oalm.1.lus • He thus agrees v.rl th Bal:tl1. Van Der :Pol . The equation. (41) •1s knom as the Brom,doh- Wagn.e,: . the9remi 1 ts advantage 1s to give the indioial aclmi t ·t;a.nee tiirough ;a complex intEigra tion 1.n the complex plane. Concerning the infinite intea-ra.J. theorem, we should :point out that while Van Der :Pol used. it to solve diffel"entie.1 . equations, Carson considered it rathe::t• as a ohe@k on operational equa:i.ions; in othe:r words, Carson did not use it to solve operational equations. It is deduoed quite naJrurally f:rom the B:tomvdch- Wagner theorem, by the,, of the the~ry 0£ residu~s. The poles of _£ __ • jJ Z(j') are g"lven by p =o and Z(p):: 0 - Z( p) = o is called the determinantal equation, Heaviside established his theoi~em in the case v1here all the roots are different,. In the case vJhe1·e there are ooinoiden.t roots, Wagner gave~ gene:l"a1 :f!Ql\'lll of the expansion theoi"eID ; but i t lisimpler then to keep the inte~·al ( 41 ) combined wt th the th$oey of J.'"esidues • Su:ppose tho..t Z(p) = o g ives the single ,roe.rt.is:. ~eaidu.e .al ~.,:n.e ,,Ql)igln,. ~~;,::o j'Z(/') j£ = " [1· £pf} - I Z ("o) - Reaidue .e/v the . roctt /" . j7_ £;,t-} • [ jfk = --lun../'= /'It · (~-j>-") jJ Z fi'J = a. (jJ - A) - · - (j> - /'1< ) b'll. t Z f?) thus Z [p) = 4 {? - /',) /' - /'1< an.d i,;,,,.,{ Z, (/J} } I' - /' /< /' :, f /< we J:i.av~ h - - - -: ( I' - 13,,.,) -(f' -l'tt _, )(;,-/'1< + 1) ---(13-?n. ) =- Z, (/'K) /'N r, J{'" :r l'k ~ {,P~) Thus: A/zL) = Jro and sinoe • A (i) = 1 2 (o) -1- :leI +- - - - + .Jt'-,.z., + I< =,,,_ ;O,c t- I< :rf /'K z ~..t) ~ £ , (4--Z) Sinoe A( t) mus,t be real, we deo.-uoe that the oomplex imaginary l"oots OOffl.$ out in pairs of oonjuge,te complex qua.nti ties. ia the na. tu:r:al .d.eorennent ls the na to.:r-al a,ngulal' ve+oei·by is the fJttnEt~li~$q. u~~~l. ~~la.r .v;el.o,~.~t;x: Heaviside establiahed his theorem tw:to-e; in his n Eleotromagnctie Theo:r:,- ", Vol oII, P~ 127 1 and tn his n Eloetrical l'ape~s ", Vol~Il'. 1 p,;226 1 aa a :f'ootr:i.ote . V~.llarta first presented an algebi..a1c indiacm.table proof',. Bromvdoh studied the ease of eJ:1 infinite number 0£ roots,. Discussion of ·the equations ( 42 ) a.v.d ( 43),. .l'-i.fter a time / =- 00 , the expenen tial term.a clisapI for a unit applied voltage. .rn."' at"". Ay state term for a 1 "' (.:I 8 • <;r<;Jvu..,;J •• •• . v: oltaJ.J"e o ·. £ • ~:wt- I is 2(1-w) - - •.. in whi0h we see that the symbelie impedance Z(p) becomes the eom:p1ex i maginary impedance Z y·w ) ; and the whole . operational cal.oulus red:uees to · the oompla,.,: imaginary cal... ou.lus. The exruuplGs (42 ) and (43) are sufficient to show the wide use of e.pplicatie:n e.£ the Carson and Brom\doh.... Wagner theor\1im1S., The diatu.vbing e.m.f. is a i:unot:lon E(t) of exry she,:pe whatever. We want the current solution i( t). First 11u~ thod.. We replace E( t} by its image: Then, it ft,lloW$ at onoei if/) = ~ ✓ _ft! feJ .;,y. J I- 2(/,) E ,,o~ /co AA/ ✓Tl(P) =? L(A) i: -p.-J~ Z(pJ o SeQOnd method, The solution is reaehed in two s t eps. As a oheck; We f'i1•$ ·t dtrtel1m.ine the indieie.l admi ttane~: Then, by the use of the superposition theorem ( Bol tzmrum-J)uh.amel integral ) • we have : tt,. . ftJ = ~;A (~J £ (f-;,,J d~ dt • Conolusion: 0 This first part indicates the eompaot form into which the develol)Illent of the whole of the operational oaleulus ~-i.ay be iut, On the. ba sis of the :&'ou:rier integral ·theorem, -we have clra.wn oonelus ions on operationa l equations. By shew... :tng then tha:t the solution of a differential equation can be reduced to the soluti<YA of an oper~tional equation, we have a pplied our eoncluaions on o~erational equa.t.ions to electrical engineeI·ing :praot 1oe:, • .A.11 the V!ell knOt'l.'U theorems of O_arson, Brom'Wieh ... Wa..s ner, and of Heaviside are r eadily obtained, with the1:t- right meaning a.n.d. their limitations. This entire procedure is opposi t~ to ·the historical development of' the opei~a.·tional ealeulus. Heaviside established his theorem, and also the meaning o:f' the opera.to:r.- p, bu"b he proceeded by intuition, and went '4\ll"Gng o£ton, • beoau.se the whole ma.tteJ:- seemed mysterious .. 1.ia.iuly, the impulse p1 :vas a most 1nclefinih1J matter, ]lf;/ opin.11111 1$ that operational ca,lcmlus $hcu.ld be ·• ·taught by :finishing wi ·t;h the Meaviside ~ansion theorem. Ths aetu.al ulld.$.EJ:t ic, methc,d in wh,ioh this thc,orem is onlf the v~x·.v atart can only lead in:to misooneeption.s. -33- Dei'eJ1lilat;ion ot th• front o:f traveling waves a~cmg power lines 'llltder qoro11a conditions. ~ ... Sr • , FIi . • 1 ! 1 (·1 -- - - - - Th• problem of the defo~tion of the front of traveling t,vaves or curges -along power lines is a very impo1·tan. t one. It i s lmo:i.m ez pe:-eimen te..11:r~- tha t, as rt. wave travels, its i'1..onit :flattens out, reduoing thus the dangei"ous mwcimum :potential gl:'adien·t which is the direot oause of f'laehover in transf'e:rmel",B a.11.d a.. o. machinery-. The a.-t-tenua.tion of th~ vmves at their £i~ont is mueh in:ve:r·sely p1:•01,ortional to the sq:u,are of the su.re;s i mpedanoe of the line , as 'liv.l.11 c,ome cut of the theory. The faoto:t•s in£luenoing these a tten,1.a.tion and dis- ·tortion are skin-effect a.ud oorona 11 Ou.:u objeo·t is to introduce these fv;otora in the theory a..Yld to Jredetemine the sri..a:pe of the waves a. t their front, as they travel along the line. We shall take, oo,rena. :tnto a.eo0'Wlt by the intreduetion of a leakag e cona.u-ett.v i ty -whioh 1.w.ll be ti•ea.ted a.a a eonstan:t. The aetual val'"iation of th..at ooe£fieient v'tl th the volte.ge , &QOO?"dil:1.g to experimental data• will be taken intG aoGO'Ullt a.£·te1·~wards in the selution. General conside~ations.~ Let us ool::1.$1.cle.r a double line of cylimlri oal c:lrou... lar 'l(Tl:i:·ea. In a transversal plane, we have fl. ser:i.es o;g lines of foroe flo1.rlng f1,oiu orJ.e c;onductor to irhe other. The com,-• pouent of the electric · field in a ·t:r.·ansversal :Plan$ will be denoted by ~ • In fa.et, the total YEH.'rcor electric field E has also a eompo..Y1ent z_ in the direction of the wires. The mv.gnetie field is in the tra:n.sv~.n·sal p1a:ne and :ts per- pendicular to the transversal lines of force. Consequently, the l?oynting vector is not parallel to the wires: this means th.at t in ·the S1J.ccessi ve tire..nsver- sal planes, at the front of the WltVa f'or iusta.uoe, the flew of e:nel"gy, traveling at the veloo:i. ty of 1>ro]?agation, is not a constant, but de..m.ps out. If we express the e\~qt,:t•io~l . ID¼a:n~\ ti Els in the c . g.s .. ele~tz·osta.tio system and the 1nat;,t'letie quantities in the c.g.s. el~clroma.pe'bie system of units, then we must introduce the ratio of tho e.m.u.• to the e.s.u. o:f charge, i. e • the velooi ty o of light• e.s waa first th~oreti eally aho m by Max;,.ve11, Then: -S: Poyating veotor ixii : veotei..ial product" ' ----- l- - ~~---------- ________ . . _ . • ~ - - - - - - - - - - - - - - '- We sh~.11 be i:nter$.~rted in the da.se 0£ a double wire line. We su:ppoae th.a. t the oondu•tors are so remote f'rom each other ·tha:t ·the ou.r~ent flowing in ea.eh wire has no effeot whatever upon th& distribution of the current in- the other. In other wGrds, the skin effect in each conductor ,1-vill be the same as if 1 t were isolated; this aJ):proximation 1s excellent;, The vectorial _Ohm's law 1.tu where E _r= c:r 'Z° - cui--rent density vector - conduotivity Since we assume that thel:'e are no eddy eu.rrents, and therefore i, are parallel to the wires. The eleetrie field E, inside 0£ the wires, is therefore purely longi- 9ti1.1a.l. The Poynting veotor, inside of the wires, ie th,:;refore purely radial too, '?his means that theJ"e is no longi-'mdinal propagation of the eleet:romagnetio :f'ieid inside of the wire. This, by the way, is a ver-r,J good proof' of the fact that the ene~Q is prop~gated along a power l1ne by the eleetromagnetio field ou.tside of the wires, and not by the eurrent insia.,., ,o f them. We may a.ls-o. as another con.sequen- ce, exi,eot to have for X or £ a. d1ffu1:1:ton equ~tion. This oorx·espE>nds to neglecting -the displacement etU"rent inside of the wire. This last assumption is excellent, because the t;i,...ansversa.1 di1ue..11.aiti>US of the wires all'"e small• and, on the other hand, the velooity of propagation is very /C . olosa to that o:f' light, 1 .. ~. ~x10 crms'" ))<ff' s~e:,. __-_··_, The Poynting veetor, inside Qf the wire, is radial and d.1reoted. towa.rd the The!'efore, the t!nergy supplying the losses in the wirei,, flows, from the outside to the inatde. C.H.Manneback, in his ;publioe.tion n Ra.tlia tion from Transmission Lines O (n.3.5, l1aroh 1923- M.I.T. Bulle- tin), develop·eq. that point very el.early. He showed, toget her ivi th Carson, that the- eleotr<>ma1.gnetie f1~ld VJaS pro- pagated from the ¢utside to.ward the inside of the wire. :Me.nnebe.ok. 1n that .way, oorreoted .Ste1nmetz 1 a theo:r,- Wbloh wa:, tha.t the field wa.s pz,opa.gated :trem the inside toward the outside of the wire. Outside of the w1res, the component of the l'oyiiting's vector, paral1el to the wires, eorreeponde to the transmitted power~ We Shi.11. now establish th• Maxwell's equa.tioJ:UJ whioh rire at; the basis 0£ ou study, and to whioh we shall apply the o:pera~icaa.1 calculus m0th0d. ' The i~vGll' s equation.a for the eleotroma..gnet1o :f'ie,ld, We shall denote: E eleotrio field Jl roagneti0 field £ speei~io induotivity ~ magne·t:1.e :permeabil:l 'by 'Ji"=£ .z eledtrio dia:plaoer::ient ,r .jf m.a.gnetio induetion ~ ct- oonduotivity /=fa dr magnetic, flux ff 0"'- dr eleotl'iO flux V gradient of V x o:u.rl of V. U vergence o:f' V-t La.plac1an 0f £ , ,- seal.air p:roduot No'te: Jeans uses ·the po la.rise. t:lon vector ? instead of the eleotrio diaplacement veotor .:5 ? = ~Jt ~X ~ , where . In my opinion, i ·t is better tc work w1 th , as it ls q:u.ite analogous to X First law. . IDhe first Ma:Dyell equation, written £or a finite cloa,.ed oi:rcro.i t. is: di /., ., -1/7 - ___,_ dt • or - Z-e ae .I dl if = ff, =/ z: dL c .. {ff, di") For a.n elementary oirou.it, by Stbkes theorem; where - is the velocity of tht,, elementary loop. u. However, this law ia t:ru.e only if zr ts mu.oh lest than the velocity o:f' :propagation of the eleotromagnetio field. But we are intereeted only in fi~ea. cirQU1 ts; and and u =o = ji3 - Vx£ If E is expressed in o.g.s. electro&tatio uni:ts j J3 - Vxl'- = - ' • ~ Jt (J) ,.vhere o is thd ratio between an e.m:unit and a.n e,e .unit The second law of Maxwe11 f or a. finite elo$ed ~i r ouit 1st 41r ~:t: + jf =_f'~d~ f ~ df = f .il, dZ -f,/£ 4 c~ {il, .Jl) or For a.n elementary oi~uit, by Stokes theoem: - - t?xX = 4-,crrE + If £ E ~~ -1:., ~r is expressed :ln <>. g. s. t>lentros t a ti o uni ts; and .Jl and o- in o.g .s. eleotromagnet:to units : vxll (r Vxff = 4-1trrc X +- -f E and tf or rrc E is the conduction current density E )E -;;::;;z Jt- is the displacement ourrent density We shall ~t1;.ck to the follew!ng ru.:ie i e1eot:r1o -- quantities sutth as £1 .E, 1J will bt express ed in e •.g.s. eleotrostatic uni ts; magnetic quantities suoh as ,.A/ Jf, 8 Oenr :tinui ty . c,qu.a.ttoas, . W • - ·-~JI ~h•,.1f ·t .''. -: :_. I . •H , . \\'here jJ ts the d:tn.f!ti ty of' $l,otrto charges. This is a l? oi saon-type equa.t:tolt., We a.re not\• tn possessio:n .o f' tour· i mpox•ta.:.tj_t relations. It vias desirable to establish them for the sake of the units as used. We conet1dtr a tl.ou.b1t line and. make the fundamental assumption tha't the tra.usve,rsa.l section of ea.oh wire is equipotential. Fu.rthelm!.ore, \Ve rtoall the. t ea.oh oonduetor 11 treated 8.$ i:f tt were a..t an infinite distance from the o... ther. In othe:r words, the action of the exterior flux is tb.e same on .all the ,lement~ of turrent inside the wire, 8J:ld may thus be ntgleoted• at lea.st ae far as the 8kln e:f'f'eet is eo11eernea.. Cb in·tegrai eq,ua.tio;n tor skin effect in parallel eonduotors of any shape of oroas St,Ction has been developed by C,.H.11Iannebaok 1. but it only applies to the steady state aud could not be of e.ny help in our problem. Al though we know that L and C are not ~eally c.,onatant, we shall see that the a.sauraption ot o.o nsidering them as suoh is vei-y good, We shall fol.low Abra.ham and Becker in their " Theorie d.er Elelttr1zt tatn .~ and e0nsider a. double l1l'le of 1nf1ni te oenduetivi ty o- ~ inside of the wire, or, 1£ x is the axis direction: From eq_uation (4h ffl. th also ff= V x A A.) :- A1 ~ 0 A ::: A • ,; where A .£?r ::, o is ·the niapetio veotor :potential ~hen, from equation (1): where f7 is the scalar potential .. •.••. • • . •.. _ JA In qu,a,si-:E Jt,~tio,~& fields, Jt =o , thus:. H = c-JA -;- J[ :z. J.z = - JA J:I- N"' == 0 17A. = J)! + ~),_ = 0 ~" (~) .).z: These two eenditlons (;) and (6) art Silff'io1<9nt to defin• L and C at eon:s.t ant.s ha.vimg e. geometrical dimensioi'l ( Abrihsm and Deoke:,. ~ae :t • p.200 ) • In t h e ac'lnU\i irob1em, however: 1 " rr <" oo , th~h ve'J!Y' 1are1;• tor otpper 2-lt The eleotroma.pe:tio field. ts not ata tiona.ey; th• ve1ooi ty of light ts, howeter, to iai.,st • ar1.d we a:,e inte .. rested in a. regitn. so nea.r ·to th• wili'es, that we h~ve thewe a dw'ble 3u.stification. in assuming a stationary field. We oonolud.e., therefGl"e, that 1-tl_ta JOs sibl~ with su.tfioient ao<m.:t"aey, to banile L aJll.4 _q a1 eons~antEJ. ex... actly as for the 5tea<ty- atate .. As we ltnow, the theoey of propagation of' ielectrioa.1 distu.rbanoee does not give a. oorreet picture of what happens near tot.he wave front. This theeey was developed by Rea... viside and Poinoa.~6. Heavit11de was the fi:ir·s·b to reach a sol:ution for the t:ransient voltage and QUJ."ront distr1\)u... tion in the case ef au initial reota.ngu.1a:r voltage wave, Hts theoq, e n t i r e l y opePatione.1,. W.1>1>0aes an i n s t a n t a neous penetl!!ation of the Q'Ul/'rent 1n the vnres, 1.e, a uni:f'om di s t r i button tI'.You.ghout theJ.:tt c r e s s s • o t t o n , On the samt hypo·thests., J?oinoa.ri developed the classical soltttion of the tel&gnpb.1st11 equations . The c1ass1oa1 solution tntroa;uees <'onstante which oan only be determined. by the, baung.a.x-y 00110.i ti on$. Tht.$ :ts.et does not ~o:rres:pen.d ti;) the Jbyaioal truth, sinee the waves ignore the terminS:1 oonditions unti1 they rea.eh the end of the line. Tht~ifitcn • the operational mE?thGd• 1n whieh the ,ivaves ' ' a.re censiderel on their trip a.long ti);e line is th,, only na.tu.ra.1 method.. ft tn.veatigation of d.istu.rban.ces. • . • • .;. , , . ' . ' , . , 'I ••· ., ., • ' . • . • ~ , • ·• . , ' We sha.11 new bring i n the skin e:f'feet. I -1c1a.J.r= Jt . I being the total cmi•:.t•ent in the section the eleotrio field at the periphery of the wires r>'i £0,.. radius e. L 8 the self intluotanoe :tor the pair of conductors The term «2 .£4 k re'\_f\laoes the known tei"m ~ dx. I " since we do not know anything about R, due to skin c:ffeot. Thus: {A.) We have also the oontinuity equation: rdt "' or Vie have a system o:f' two equa.tiona (A) and (B'), w1 th three unknowns: Y • I - and E • The additional equation vdll be given by the study of the eleatromagn.etio field, by the use of 1ia.J..-well 1 s equations. We shall establish an operational :t. elation between 9 the tengential eleetrlo field and the total ourrent in the section; it wtll be the third e-tion. 1• z. 2, There are no free electric charges. and /" are supposed constant. t Vx E - I"- (SJ (: (J) v. H = o (S) We have the general. veetoria.1 relation: If A denotes ei thar E or Ji. v.zE = 4- 1r rr µ Jl" + / tV"]{- Jt Jt-l J.il + --L..£. '-1.. J)? - ~ 4-7ro--µ. - / £f'- ".t .: t7. A =- o J.z.£ Ji '~ Ji-l The terms- 1n Jt ~ are cUf:ru.,ion terms. -- -- - .The tel?lllS in ;;.: ar& propagation terms. 'cJt-l -_ -__- • _- -__ -- -- -- :.. (.9) We h av e a l:nl~ nr q)a.t.~tion,. C'f: · .;;;x; ,.. . ) . ., i ...... ~ . . 2 • In.sidf:> ef .'tthe wires, ~ + o Thi s is w.1 exd0lle11t approximation, as oan be aho,m by wha.·t follovm, Let us as$Ume fo!· E and j.wt: oscillation .E JE - . Jt =r.J- w .!. J:l @. w / ~I oiss o1d.al , the.nt ,:Ju>t oa- / J~E / - l~l=w"}Ej 4;4 ~ JE I and TC~ The condition that th-e propagation term be smaller that the diffusion term. is: For conduo~Ol!'S, E , though unkno"Wltl., is sme.ll ~ a.t industx-ial frequencies, and even voiaa freq:u.enoies, 1s less that er .J ~ ,. 1000 couduGti vi ty o:f' conductor, is always large. FQr copper, the resistivity in e.m.u. is 1800 . I /$00 and =3.I'+- X/0 17 (' oonduotivity in e.s,u.) For iron,. the :ceaiati vl t y in e.m.u. is 10 000 .. ¥ .· o ilf • I ir 7- Th.us o- ~o( in e.m,u.) ~ /-0'-000 and o--'"~ - ~. 10 /0000 = •J x /o ' 7 ., For . copp~l2', i w which ia the oaee ( oom\uoti vi ty in e. a. u,) f tJ /8 should be tmtoh less thtui 6. '1-s x 10 < <;; 4- 'IC r:r <. ,,: ./. 4- n: rr-c. :: I. /3 X /o /B ~his justifies our approximation. We neg+eot the displa cement current by pu.t'liirLt!5 simply £,.. o , wherever it ma.y appear. Thus: -~Jzt Jll Jt !!!he physical meaning of these equations is that if' any modification oooura a t ~ given point, its effeots are felt at the ver-iJ same instant a t ~ other point. Diffusion phenomenon :ta .therefore a partioular ou.sa of propaet:ton . with .&lf :!.in~1n:t te velooiJz. ~he general formula. for the Laplaoian of a veotor in ourvilinea.r coordinates, e.p:plied ·to the oyl:tndrioal eoordinates, is: \7.:'f = _, l_ f 2 )A) + _J_ j)l- + J:i z Jt l • J-2.. tz.-l J""'.2 Jx.z. Now, we suppoae a oylindr1oa.l symmet:cy. Dettt~tion cf £ and Jl inside ol the wire. The r$iiu.l"t!. wl::.-e f .1>r th• ~rent is s-u.pposed to be t'll. t an 1nfi.n.i. te d1ata.uce. ~ t,vc MaxtV'el1' s ' v r- lt = 4- 7r r,-- equa.t:tona are : Jt- ~ e + .£ Jt Jf ~ Let ua consider the folloWing closed path. consisting ef the eiroles of radii rand r ar. plus two elements of len,gth dr. We apply Stokes' theortm to ·thll.·t path. Tha positive normal t o this surfaoe points toward ue , in accorda.noe •rd th the direotion of oirou.lation along the path. The given - direction or H corr$a~onda also to a current pointing toward us. Therefore; - ll:..uz. + {1-l ,.!h -,. ::: or (4--11:_ o--= E + / ;i{ (,;n k) ./4.] = V'I' Jl. j., ; ; ).,; .,,,,_ .d.. _£_ 2_ ;;_ A5-\ =· 4-;r (1-c 7:7 + _!,_ )£ '2 d 1 / 'J ••• 1/ ~ • Jt Another closed. pa.th is aho\V!l here ♦. The supposed direction :f01• if i mplies a mag.netio field into the paper. In order that t he positive nol"1l,lal to the cir- cuit be in the same direo- - tion as H, we have to eon- side~ the. direction of oiroula.tion as shown by ai..row$ mt t he sketch• f' Z ..:C = _/r,, E. d-:r The equa t1$ls • ( 11 ) ani ( 1 2 ) ce.n then be t ~ sf o:t"med / as follows: • ..!.. -'2 or 1-. (1 dt~) = )1. _f_ l_ z. ) 2. €l.lso, by ( 11 h Thus , by differ entiation with respect tQ~. J:1 JJl _ ...!.... 1l :: ~,ccrc. 'JE +- ~ }_ 2.t d'l.. , Jt ) 2. I ~ /_ ~]{) I ";)JC ~ )"•J[ z5i: {I],, h - 2.: H =- 4-7C~~a""t: + ~" "')i< ";J 4-H +- L d 2.. ~ 0..,. .I. 2. (~£) ~~ · Negl eottng the displacement ourr en~ inside of t he wires, OU1· e(Ua.tiona be()11muas _f_ -.. ~ (~ Jl:) = 4-,r rr ,,t.. ~ i.. d'l. / dE dt 1- /~ d2. Jl[) - _!_ I{ =- ~ 7l A. JJ[ I c>t _!_ ~ ;)~ ( '• (J'"' z.t (tj) (14-) Lot A be the co11duo·tion current density at a d:ieta.noe r :from the oenter, ,\. at the pe:r·iphery. We ahal1 neglect the displacement current density., 1.e., we shall oon&ider the only)... density. We oons1der the finite cirouit between the radii I ' I~~~ law: I I -r:r ' r being the magnetic £lux: per -unit lenei.h between the radii rand a, We :have also: . (io) where the minus sign is due to the f'aot that if r increases. r decreases. N'ow let";i ·'.;b& the total current i.n the section of radius r : L: : ,,\, .tJCz /4 or ·c::, -if' : <lt'l.. A (t7) Fu.l"tlle.rmore: (18) and The five equations (15) up tc {19) oontain six para.me tars : E • }[ ' ~ , :r ' r . and -2.. t It is therefore :possible to eliminate rour of' them and from ( 16 ): from ( 1.5) thus identical \rl. th ( 1J). To find H, we proceed as follows: d~· )A ,,_ ~ '"' ,,tf- er d 1- but :::: ..3:...K -Z ~ thus J }[ 12,. . ere JX -:;:- ~t = -1,.~U:>~ or ';)Jl - C . (.. d_E ~-F·~ identical with . ( 1'2). We have al.sol.~= VJe must eva lv.a ·te iclonti eal with ( ·14). Uc a:r·rive t hus a t ·the same 0q1.u1tio:n s by a com:vletel.y diff'erent method. To clater..niue E and E, it is not necessary ·to solve both equations ( 13) and ( 14 L Vie ce..1:1 solve for J, for inrd,ance, and determ.ine !I fro1n the eque:tion ( 12). Solution o:r the elec-tr-io field eq:v..a tion . We have: This is a B~ssel-t;r;g~ e9:!:1A~i,on of ~eroth ord.er. tle use now the symbolio r ep:r osentation /' ==- d~ It is ouatomary, i11 all skin ef'f'eot problems, to introduce the qua...~t;ty; then: +- ~ ~ V4 7t r 1• = ~I' E La·t X be a new variable related to r by: X ==J',:,, and let : Y= ~"' Vf' °' I/; so that: X =j y Ou1· eqaa:t;ion become s : -;l-.e + ..!_ . "iJE ~ X"' X + E == o ~X Thia in a Beas el e(lmtion u..11der its usual :form~ The most general solution of this eg:u_ation is: whex·e A and B are oonstaiits of in:hegra-tion and I and .7l: a.:t""e Beas~l funo tic>n$ of the zeroth order and. fiz-s -t and s econd ~tb1d respeot:l.V'ely, The eleotric fiel d is a oontinu.ou.s function of r or X, and sinoe 'J< (.X) is discontinuous a t X =o, i t :follows tJ:1..at .R ~ o , ao that: Rolation between E_ e.nd I " The eleotrlo field E is thu.s exi,rasaecl by the operatioL'lal equ.ation (22). The -equ.ation ( 12. ) can be v,rritten: LatX=-X4 andY=Y for l"': a, i.e. at the periphery,. 4 We know that : .J[ 2 4 .: Z , where I 1a the t;otal ou.rrent ~ in -the seotion, Thus: 1 I :f= ~ - ~ . 1'ol\fj,.', A. T0 (X) ..t 1413 (/ : 4- I ~ _ .t '!fCJ-a...z~ A or , :,;- {XG..) X, • Let R denote the d.o. reaista:noe per unit length of tha double line, the two wires being plaoed in aeries: Notice: Before going any further, we may oheok on the dimension.a of this la.st expretssion (Units-Jea.na.p,.530) r~J = [ .l r~'.L-'] = [r-J [X:-~•'t;:} fX:~ =[L' ,['r, r·J., I [.Z-] = [.M.~ 1 f 77-j in e,m.u. in e.m. u . in e,m.u, [~]: [Lr·'] 1 then:[~. X"". J;CX:.J. xl = [T~/tt.-i:Li r·' = [AAt1 -t r·'l ~ ~'(X~) j i r-, ] J'L J wh1oh re11reser.rts [ E~] in e. s .1;1.. of potential 1,1er unit length. This ahotw the consistenoy o:f ·fjha ro:1:'ffiula ( 24) 1n wh1oh ma£,netio qu.anti ties are expressed. in ma.&."ne~i..c uni ~•t and eleotrio qu:a.nti tiefl 1n el.e otrtc -un1 ts. VJe have the general algebra.io relation: ..4. dx. [x-- ~.,. (X)l'J x-- Y (X) 4- ~(X) (X): - :r (X) = - M ♦I 1 thus := ~ dx and • ..t E_....: $. X X , .z, 4 _ 1 J:(X.J J; {X.J In the n Fu:nktionentafeln" of E. Ja...h..Ylke e.n.d F. Emde, we have a general formula: X / _,...;:. Ci«J = ~ .( r-.:o<J (.ts-) ,s'(- .:Y) = I + , I y • I. 8 By dividing~ by J" • we easily derive ·the new series 41 I We notice alao that: r /,;,,. :r ClY) .::; r➔ v~ .Y • i=~= o;, and 7l' or Variation of the dif'ferenoe of potential V between the \vireu,4 Using the symbolio operator p, we have now: We have thus three eqa.ations and three unknowns, a.nl proceed to solve for v. But, if Vis expressed in volts, I in a~peres, and so on, we mu.st expre&Js ..2C in volts per em. , whioh oorres- ~ :ponds to exp1·easing B. in ohms. Then; .t£.. •l-f +§-""°''I},+?, (#JI in voltt1 om Inoidentally, (4°' ~} is dimensionless &.ml rm:y "be evaluated i n ~ system of units. Thus : d -z! = [;,.z +-;:-- ! ~ + 40(VF + T,(VFJ)z, _ di: • [ Cr + ~ 7y 'J d.:x. and d<Y: d.a:-t ffl.th t~(Y, v-,~_ O y ,.a-,(r.l + (:.2 B) » f- i-""' vj' 'z; (v} + + - Let us introduoe -tr= - I v~c , (.i.J) As we shall see , . v itit - the velooi ty of p~opa.gation of the waves along ·lih e line. Conse(fJ.ently: - ..5 9- The equation ( 30) ma.y be trru1si'orn1e<l into: wb.ero r is &"1J_p:poscd to be oonertm.rt. It is logical t o te.ke % = o , sb1oe ·che voltage ·will not ir1.0raase inde:f:"initely· ·with larg e :ic,. 11: i s an v..n...½now:a flunotion of time, say E(t), t-~~ £ {i) {.I~ then: y: -~ -tx It is )J'u.J?l)Osely th~rt we write £ ,E(i) and 110-t l'aJ,£ • ainoa f -f.x. 1:j:, · is m1. o:pe;rj.ntt operating on ·the time function Ef:i). Aoco:rdi:ng to tho va1,.,1.e of f - : Let : , _f'{j'-x - ;1~ :1(. L t: , we \'.rt te : _ -x ~ (._f_l £ ,r;;-~ . £ E,(I) ,.. ~ - "' ::t;, ( E (i) #J £ ff) 'being, o:f. oourse , 8.11 unk".l!.O\'tl'fl ftu1c·tion o:t' t:trae" Y.' P .A-X. f - -;=-~ : -;- . £ _ ,rv;; X . £; ff) according te a well know <r;2~r:a.tio~al ,tranrs£~~- . fo:vm.u .~ ( See itot e 2 ),, V = f. -~ ~ z - I tj' -x, £I ('1-I_ ~) ~ Thus : In the note 3, we prove the i'ollov.t:1.11.g operational velooi t, . of I>l"OPt¾::.~ ti.on. al.Qng the line, In the el!:Pl'".e @1ion ( p,5), eve~tl>S.~'- ;1$ k.noivu. but the time fu..notion i; (iJ , But, l ,zro -x # o : Y: _::_ E ft) 1: -~d~ v,r· I .J = .£, If) 0 From ( 34), a.t -x ~ o , Y: l'I)• Thu.a t E,ff) = E(i-) Our eJcpres$iOn ( 35) is en·tirely dete:r,mined. The con.st&.n.ts /1 , J' ~d v oa.n be eompttted and l; /.e} is the applied. voltt:\ge flll'lotion a.t the beginni~ of ·bhe line at i = o ., For any type of vplta.go applied at x =o• the formu.la (35) gives us the mea.ns of evaluating the dif:f'e;renoe of potentilll.l bet ween th~ wires as a i'u.nction o:f time ate. gtven point x. -61- Discussion of tha vol tatse equ.s,tion ( 3.5). rla shtii.ll diso1.uss i'irst the vol te.ge equ.e,·tion ror a :..:octo..112,'l.1.lt:...:r ~.ppl:i.ed volt§l,ge i..i-t the begi:un.ing o:f the line. This is the moat 1.m.i'avorable oase; tlrus • ·the oonolusions e.t whioh ~Ne shall arrive vdll put u..s on th0 sa±~e sir.le in a.es:tgn:tng vro·tecti ve appe.ra tu.a. - For i ~ ~ , V:: 0 • Therefore, the a.iff.erenoe of pote:n·tial at the front of the wave i s zero. 1lhia :property i s q11i te general e..na. does not depend on .,che special t~-pe of applied voltage ~;tJ. rfe have established the :fo:t"m.U.la ( .35 ) o E the basis tl1.~-t; ~Ghe skin ei':f'eot in one \vire 11epenG.s 011 the v1ire its elf, fore :1 -tho f'omru.la ( 3.5) 1:.old.s £0:f· o, tl1.ree phase line VJhere v· a.1.1.~ ,E, (i) ar..~ ..e~the,x· ...v~pos ml voltae,e.s o:t· l)llt',s.e ·v-olte,ge,. lfow i however 1 L 221cl C are the i:i:1ct1J.st:r·ial self indu.o- tc.noe 0.11.d es.:oaci ta.nee ( note 4). In pa:rticulm: » we r ecall . .I . ..., that the sell i :nductan.oe of ·the three plw~ac line is exactly half of' that o:f.' the doo.ble wire line. a 1.1.nifon.u oui·rant; in -------· we know tha;t;, in the case of a double vt.lre line : J?o:e distribution of• the t-Jires, ~ J; =; ./4..- + 4 / -"' .e.... _!!_ -& '2. /-\: 11er om. of length, where ..-~a. l'--c. ure the ir.:terxw.l r-..nct e:1s:.te:t:"J1al ncrn1es~l;ili ty. Tho first tenn i.t~ oxrn t o tho self' inclu.otio"t1 i n. l)oth 1-vires, tho 1-:10cw:nd ·t;o:i:."Ll. i tJ introdu.cecl b~r the rrnli' i:nduotion 1H:ltween the wires ( note 4 ). -62- Conseq:u.entq, we may doubt the acou.raoy of this value of L when there 11 a pronounced ekin e:f':f'eot; even more, doubt that L be constant at all, But, 1:f' we look bac,k at the wa:y- 1n which we obtained our equation (A), we see that Lis the.self induotanoe for AA. 1 BB', 1.e. •xteri.orly to the wires. Therefore, in this oa.se : ~: 4-Lt. ~ 1J . . .. . I~ ~ For iron, where jL~ is very larg•, this remark has 1ts Ml 1mportanoe. Defo:t-mation Qr an ini t1a11y reota!J.8'Ular wave .. This is a partieula.r oase of our equation (3.5), in whioh EI /f'/ is a oonata.nt, aay r0 V = £ -~ ~ r:. Vir J I ..:!._ cc . Therefore : ~ - 2 <dz. _la: or V-2 ~- i -ti"' [ 1 .1V~'!, ( ,!,,. ,)11 (~ 6) ~vt - ~ ~ The ltal\~er 13 and ,,f' are, the smaller V is at a given point. We see that d" 1s direotly proportional to VF , i.e •. the surrae admi,tt;an<)e. ~hus, ✓ ~11 be la.rger for a oable ·than for an aerial line. Consequently. the front of a wave in a. eable nattens dovm much more quiokly that along an aerial line. Let : ,v-t • x :: l j p ,/.?C z j t hen: A_. • V~ria~ic>n o.t V: -~ ,f'un,ti_Oll ,of ,time -~ t 8:- §iV~l't ' f • u • ' - . ) . . • t ~ ' ·• •' . . . I ·- and : v;? a..11.d At a. given point x, the_~l"a.dient of potential is ~a...~imtUn (f ~~.z) •~CQI)d.$ aft,e~ 5t~e 1~1?i.!€!~ wave has s~~<pk J. I - - . J.. . the point. J. , .. . l . .. , L . ... t , . . !! . . . ~ ii! .. - l . _. . _ t ... t ..1 .. I r ._ . . ,1.1. T . X C a. -I.I J~l:to:tti,,, this time is : -;:-- • -t..::-.. --z- • -;::. . rr ci'l" ( C, L, c,. , j'- in e .m. u, a.e we d.eeided ) . The :E'a.:.rthe~ we ai11e e.long the line, iihe longer that time ta; this time inc~eaa~s with the square of the dista.n.... ee to the origin.. !'Che oorresp~ing muimum slope ts then: I ~o the ori~. l'leeee that (jJ}_,_1.e :proportional to ,,r~ , or to {- 1 the 8lope is therefore mu.oh smaller for a ea.ble than for a.n aerial line. The ve.lue of V, .wh~ the .sl~J?e .is ~JD.um, ·1s: From. the ex:pression for cir, we see· that, at time i • :, o:t· 1: o : [of!!:.) =~ 1 ~:: £-;4'< 1 1-l- L -,L_l ~ ~' ) : o JJ ( note .5 ) f;,c X Thus, when the volte.go . 'l.l!JaVe s-t;rikes the point x, the o :l.n.i ti_a.l val:µ.es of the di:f'feronoe of potential V and of the ini tia.1 slope /_1~1 a:ve ~ero. - .. . l~ili~~ · ,.,. Thia h~lds tor a.ny type o~ wave. In the oa.se we aotu.ally consider, we have the diagrams ; - B. Va.i"iation o:f' V along the line at a given time t .. The ealcula;ti ons are aomewha t longer• Since, now., t is eonsta11t : j -=- - d-,c V: -V:, .:-;'" [ t - (ii" ./~' £ -i<d:,.) :/!: -V:,;1.:-i<"[, = - 0 -<of\ J .,_ v,, f -i4"v; but thua and FrO"'JJ. thi s eXl):ression., we oheck easily that: dY=- 0 d.~ We must, at -::r ::r o therefore, have a maximum g1-adient somewhere -66- between -:x -= o ✓.:t: : v-t • and l Unfortuna.j"ely, the exp:r:ession f'or LY is too int.rt... dx..:. ca te to be handled sh.1.ply" IIoweve1~ • this pa.rt B .o f the study 0£ a rectangu.la.r· applied voltage is less important, sinoe what matt ers is to know how V varies in function of time a·b a given point x, as vms £ound out in the part A, We have : et{ = V4-Tr j'- .,J' = <1- .!!. VC • "-°' ~ .l.. 'rr.;, - ~J £ V.£' I' ~ ~ 8 VL( li'or a double 1,nre line : I 'Ji L 4- + c2, L = 4- -ta. .i, C om/ om. 4.. C :::: I om/ om. 4.- and a. c_ I ~~ ..JL ct.. I but now~= --:rand A <r1Ca. /- has the same value.as for the double wire line. Thia, 0£ oou.r~e, could be expected from a physical point of view, as the steady state va.lue of the yol tage must be the same~ In the same -.vay, there ia no oha.nge fo:r J ~ Let tu, consider ,e. three phase aerial line ; D ~ 10 meter let a, 1 cm. I /Soo Resistance: -~ = 1800 _1£ = ....!,_x XIO,- ~ i't /0~ ~ or Jl ::- CJ.os-7~ 6bm/km. Self induotanoe : _£ = ..J ~ _!_ " lo -.:1 . 4 L ::: ~ '/.-2,3" 1J lo"o /o-{ /o s-::?or L ~ /, ~S x X I 3 Henry/ km. f:"e£ ~ lo s- I Ct-rL " g xlo" · - Ca.pe.e:t ta.nee : /" /o- /.I~ C= o,8c~ X 10-8 Faradj'km. '- = ,2-,;..t. - 3-X3 - X ,jX./0 --11 Leakage oond;u.otanoe oo:i:-reapo:nding to oo:eone. losses ; an average value is Velocity of waves Km/ aeo, Suge impedance Evo.lu.a:bio.n of/3 : A ;- = c, os-7t,.. 8 I f I x 4-tS ( I - /$ .8 f ~{l~xio-•J +(..tc.7~ x 10 ) -J -,.. Evaluation of j' : X v~Jr X - 1 -- /Soo Time for the wave to reaoh the m$Jdm:Um potentia.1 gradient a.ft~n: striking the point x =100 km. : Maximum voltage gradient: -- -rr f . '( - sf X/0 -i0 0 . fj.,/4,. "· 0 84-- ~ )-'~C.. 0, !JJ6.i thus Du.ring 1 f"- sea, the wave tr-a.vela 300 meter• and : II y; If' ~ = /ODO l(y , then ; /_ efr ) = .sti-J Kv { ~t: /4,,~ -mefef\. This value, :;6.7 Kvolt 11.eter, ta the critical . vol·he.ge gradient ..... at e. diste.noe 0£ 100 km. tor a 1GOO Xv., ~ . initial impulse. j,4---seo. G:enerai Case. Deformation of. e., vm.ve o:t general Our e(!'J.a tion. ( 3.5) proved to be in a convenient foroi fol" a rectangular ·wave. However, in ·bhe general oa.se, the integration that has to be oa:r·ried oi.r!i from a given value ·to infinity is, of course, a graphioal impoas:i.bility. We sl1D.ll dis<ro..as the general equation {35) and. Let; For 2. 0 < 2 < cc Thia 001·responds to the pbysioal truth, since, at ·liime,t, only ·the values of£I :for \Vhioh ·the argument lies between o and ( t - 2£. ) oan af'feo·t; ·the voltage, ~ -'10- Thia proves 1n a new v.Jay that v WJ~at be the velootty thus z =,, 2 0 For z # - .. I "-o ·= 00 '"' ~o - -:1. Vr- ~ ,,,r«. 0 After an easy trmis:C'orrn.a-t:Lo:.n, wo find : a I Y~ ....:L i -13 Jz[~l(u,t-u..z)1 f;- ~ k v~ I 4- o ) '"-.:, 0 Under this :Eorm.. the ;.1.p:.•e.phioal integration ce.n be perf'orD:1ed in a, moat convenient vmy. Before the integration., I 1 of the ourves ~ - ""::t ""' oa.n: be oaloulatedo • 2. I I -;:z:r t For -~ ---u... o o Let this ftl:notion be f(u). -71- It is convenient to build the following table: I -;;:::: I I /ft.)= },,l f. - ~ ---:r u i 0.25 16 0.112 1 .. 7<-J2 0.50 4 o.. 0H~2 - o.072e 1. 73 o. ·16b 0,.279 1 . , 7.6(1 l) _.. ;; u 0,.J68 Cme..x) 2,.00· 0.25 o. 784 r' 1 °6 U " ,I i:; ./ • uU 0.04 10.00 0 .. 01 0. 963 0.99 0 .. 01 50.00 0.0004 1 0.0004 . 1 DO .. 00 0oU001 1 Ov0001 ,-, .•, 0 .. 0.36,5 c.~ I - - - - - - + - - - - , - - - - - . - - - - ' - - - - - , - - - - - , - - - - - , ~~---..&,= C In order to see wha. t is the 01·der of magnitude of giving the range of 1nteg1.'8.tion, let us take the line we already oonside:red, We found: ,/:: ..t, s, "10 At a distu.noe x-== 100 km t also ; ::., = --V loo 300 0 O o -o ,,---. ~=~ J~ ::: ./. s, xio-4- y;e~· thus u.0 = 4 Vi-~ 1'«. .t"I..IO" :: -?. 8'] ~ /0 3 Vf-333° = 6 -.J3Vt-333 . i,.:here t is expressed :tn rd.oroseoond.s. Let ~ be the time after which the waYe -ha.a ~aohed the point x: ~ 1'00 kra, i .e#· r-I a i - ;;..; "o =- {;.:J.3 '/ t; • In all praotieal OF~aes whe~e lightning surges al"e oonoerned, it will be quite au.£:t'ioient to take va.luea .0£ up to 30jLaeo, sinoe after that time; the w.lue of the ta.11 has oome down to less than half its pea.Jr value. -i• tr,; I lllf'- 13filC - u~O . . 1 6. 5)3 4 1.3. EG 10 21 .. 90 ,.,.,.. G.V .31. 00 30 38. 00 Within this :iw.nga of values ot u., the graph:1.oal intetration oan be oarrled out for a.~y parti011lar case. Lightnine Surges. 0 f: I All the lightning rru..rges ind1,,1.eed on lines h$,VO a very oloae sha.;pe to the ·double e!lCPonent1a.l vm.ve : -e: E, (t-a.t_ € - br) {~I) oom.bine.tion velues, either real or complex. this wave tal~es v.11 the possible By g:I.ving to a a.nd b different pre.otica.1 sli..apes. I ·t is thus well worth to investigate the behe. v1or of such a wave. From an osoillogram, the equation oan be obtained. by reading ~""' , ~ and ~ ~ In BewleyT a 11 T±·aveli:r.i.g Waves n, we see how OU.l'"V&S I, 4:. ?" , .£,,,."-" taJ.1.d E, of I ( ~ a..lld ow1 be dre.vm in funcr"Gion a~ ! in /L. sec.,) ---a.i, A,. .A:ny \'iiS.Ve is defined as z,m~I" I 1-.i. I~ ' -the sign corresponding to the polai·i ty of' the wa.ve. Aa an exu.r::ple, let us find the equation of the wave ( osoillogrwn ) : /ooo l(v/1,0/.t I/+- . We have thus : The g:t"a:ph gi i.res theµ :. i- .,/I t:, J ..J. :: ~ : ,, 0 ..!..=J8.S 4.- af :::: o./,J,i.. I l:,mA!f. ·thus a. .: - - = o. 8S'-t ~ The P,hysice.l reason of the . double . exporientiaJ. in • , the lightning waves. Ji.. • cloud hw.1ging over s. line cre~.tes an electric, field whioh nrf!~Y r eaoh 50 Kvol ts per f'oot o:f' height. !\1his field induoes charges on the different wiree; these charges may of oona.u.c to i·s . When the cloud is discharged by dirac-'i; sti"'oke I two wu.ves are sent in 01n>osite direotion.s. IIOYi.rever, the cloud cloes not clisolw,rge ms-tew1ta.neou.sly, bu:t aeaord.ing to an exponential disoharea tu.notion. It oa.n. take up to '.OOft-seo. to disoharge the cloud to 9.5% of its ini tia.1 ve,lue" By applit}a,tion. of ·l;he superposition theorem., it ea.n then be sho\vn that the two waves sent in opposi·te dix-eetiollS on the line he.ve ·the tlouble exponential shape. { See Bewley : 11 Tr$..usmiasion Lines u )• Deformation of the double e:q;ionentia.l wave~ We jv.st have determined the shape of a lightning 1;1, ave. Le-t ua investigr~to now its G,:ttenua.,..;1on 1 as it travels along the line. Within the range of integration, this :f'unotion is finite and the graphical integr~tion is easy to carry out. A£·te:.r an inf'1ni te ·time , "c, =- ~ , arz.d redu.oes to zero. Tnu.s V = o- f'o,: t = Q;> Conelu.ston. the ou.:i.--ve , ~ -A("/-"-~) as oou.ld be expeotea. Our voltage eque.tion (3,5}, modified into the form ( 40 ) can thus solve the problem of the attenua.tion and distortion of a. wave of any shape, under corona oond1t1ons, wh~n G is constant. We shall mee very soon now how the variation of G oomee in, It is a known •faot that above the corona cri tioa.l voltage, the \'IJ8.Ves a·ttenuate rapidly until their voltage :falls down to the ooroha starting voltage. Fu.rther damping 'beJcee pla.oe mu.oh more slowly. Assum.1118 a line without distortion and having thus a. mere dari1ping, we oaa find out the rate of ohange of the eleotroma.gnetio energy, a.nd put it equal to the rate o:f change of the enera absorbed by corona losses. Aooording then to the assumption made for the losses, di:ff'erent la,vs can be :f'-OUD.d :ror the attenuation of vol·ba.ge waves. 2hese well knovm.. laws have been established by Skilling, J?eeks, o.n,d Foust and Menger, They · are. generally well ·suf:eioient when attenuation only ts concerned. However, if' the rea1 behavior o:f' the \'!/8.Ve, attenuated and distorted, :is desired, then the general equation (40) must be used, Corona d.istortion only 1-vaa considered by E.W. Boehne (A.I.E.E. Trans. Vol.5Q, p • .5.58 ) • His theOl'"Y of d'.lffe:re:u-i:i velooi ties for ·the i.vave in funotion of corona seema very doubtful. It may serve as an explan.ation; it oa:n not lead to any computable rehlt. If G is treated a.1 a oonsta...11t, 1 t ·will have the ave~e.ge value : 0.1 x 10 ... 6 mhos per 1ml. -77- Ground wires. Thoi:r action ia to reduee the peak value of the lightning sur~es on the power lines. The proteotive ratio that they introduce can be oaloulatecl ( See Bewley ) • The :i:·edttoed voltage y,a.ves are then attenuated and distorted .as we have established., Varia.tion of C¼. When oorona takes place, the' leakage oonduotanoe is not a oonatant ~ inore, In the general equation ( 35) or ( 40), G comes only in the dam.ping :faotor /,J It . Tho constant ✓ is in- dependent of G, therefore the spaee.... time integral: ./4 ~ tit co ~ ,/i. z. .: EI Il.f_ ~:::_ _ _!!:_)idz. 4-~~ 2 l omi be expressed aa a .: Vi- <f:/,,,.. function f(t,x) and can be evaluated independently of' G• .z. In all oases the eorona losses a.re given by (r-Y-. - h-perimente.lly, the follo\il,ing laws have been found: er=' .A (v-Y) ~ .2 'r°Y.: .B CY'-Yc) ( Sl:.:1lling )• G:A :Y-~ ( Peeks ) G = 3 (1- -v-~ Y>V: f ).z Y>Y <: The fil"st tex-m of Thu.s : / where a /-3 is e. eonstsmt, ae.y /1, . /1, + 'f {rj f{r) ~ ~.L yr .A. C V-~ y..: Y >~ : - l V! :s (i - ; )~ v- >~ • .!. .z Thus : i VC.£ •. kY ~x '-"f'{t:} • Yt =(~~) . (Vin Kv, t in)'8eo.} C, o.t "lo.. ~/(~ 0,/'f- ,uo -~ For a oerta.in gi\t'en .point. x, the function f(t,x) will look as follows E fig.1 ) , On the. other ha.nd, the left hand member of the upper eq:iiation vi.rill h &:ve .t:P,e . aspeot o:f' the £igu.re 2 Qr 2', a.ooord.ing to the ad.opted law, Then, at a given time t-, , we find, on fig.1, the ve.lue of f (1 / «) , The :tigu.re 2 , or 2 1, gi vea then a. t once the value of .the voita.ge V, at the considered time. We see olea.rly how distortion comes in above the corona critical voltage. The variation of f(Y-) is olea.rly' ahown in the figures 3,4 and .5. 0 \r V I rt,, 'l:J. " f(vJ I-·-z--Vc / ,ry_ A 14-Y- I '- V .LyTx V Genero.1 solution 0£ the telegraphists' equ.ations We ~ust have aeen . ho,v eusy it is to ~ea.oh a. solution v1hen G varies, from. the solution for G oonstant. We oo'11d treat ou1· solution (p,5) as we did., because G influenoes 2nly / , and not ,J'. However• J.et ua try to obtain a most general solu... tion by the opera:bional treatment of the ·telegra:phista' equations. ,)V- = ..:E + .l.;,z- - - ~ JIJC - d~ 1r # • c/:' Y + o- v If a solution 0£ such a13.- equation oa.n be round, its operational -treatment ¥Jill give the a!'.!.Swer, For example, let : G. KY , then : w1 th - thus• d.l'Ji ::r • d,x' and l t .z_ [/!;,) + L/' C;, ,m..: = [f,(i,J + Lj'] X but a ·t i = oo • -'c;hus the constant 1.s zero . The solution of ·'Ghis operational fonrro.la oan. be expressed in terms of e. series, but is very complicate. The d.ireot solution :f'rom the tele~a:t>hista' eq'ltations is to be a.void.ed. Besides ., the integ:ra·tion is i:mpos sible 1:f any other law of va.1~a. tio:n of G ia oonaidex-ed. The ooncl.usion is that the method of va.r1ation _, of G in the so'.l.ution for G constant, i~ the on!Y one . Additional notes. Note 1 ,,. -••·· We he.ve the equation (30) : ~ ;., ~: : ; - "'" or: n : ( (l. . I -1- ,r '3. , 1' rlfJJ , :,l a . ~°' , c;;-1L' +- ;:r· T + ~T 'ff + jI" I R "'°' -ra er -' , c;a "'°' ::r-~ I+ iz''ff +l;;y ~ TJT + .2-ZC-;;F + + ~B. ,...!_ + - ' ? 1~ l + 4-.l( /'4. ;,L I (~} .iL P ;_!_) 14.[, C I { ~ The radical oan be e:x:pa.nded in v. series by the f orn:i.ula : VI -,. ~ = 1 -,.. ./- a - T x z. - - - - - - - Independent term i J -84- Vie she,11, folloiv:t.ng Dr. Buaht s teJd;book, evaluate ·hhe ope:ra.tio:nal foxw.u.a : £ - -f;, f /iJ The operand may be expanded in a series: the fu...YLoti o:n alon@; the x..-a.xia, 1 1, I Ii I 0 l'iote 3, :II I The obJeot 1a to find the original time ±'unotion ot the g1v,m imlllge I r r Vr'I #J ,c We shall use one ot the several :r.onns of the SU.l)tr• where A{ t) is the value of i { t ) when f( t') == 1 .. i( t) oa.n be expressed by a.n operatiov..al equation in which the image is f( t) ope1'ted on 'by e.. oerta.in £unotion i (IJ -;- Ff/.,) /Ir) F(p), 1.e. : A ffJ -f r(!.,J Conaequ.ent1y : T(;,J.A(I) :-'- fftJ. A(<,) -i-/51 '(;\)/Ii-.:\) dA 0 A /t) -;- F{t) where ; Let now : Ff;,) == i _-if-xi/ii"" In all operational textbook$, the ~olution for F( p) 1$ g1 Veil : "Tj Tit)=~ _-fx ~ l~ J -t I.: 7 l-~.:1irJ =- I - VT 0 )',x. f f .. 2, ~ £ d2 = Att) and d,\ As £a,i, a.s the limits ax•e conoerned : The prtnoil)le ct wpe.-poai tion allov~rs us to write . the ao called Mo.x.wel1 equations for a. e,ystem of oonduotor, • . . . . ,. -~ -· -··. ~ ~ -~ t ....... V.= 4,,1, +4,1.o/:z. + --- - ~~fl..1->L. ~- =- ~, '1, + a...u. Cf.t + - - - - - + ~-- '7,,._ I ' ' V: potential q : oha.rge a-~ i e<1effioient of Jlotential ( having only a j geometrical dimension ) . All a.'s are positive. The equations I may be solved for the charges: "t, =~ Y +c V + - - - - - + c. Y II I I~ ~ 1-n. ,n.. ][_ ojK l ooef:t1oient of induotion ( negative ) o)j .: ~~effioient o:r ~aiaci-tz ( positive ) also ; cIt + c14_ + - - - - + cI fl. >o By successive groundings of the different eonduotors, the e 1 s are determined quite easily ex;perimentallz. The a's, on the other band, are readily calculated.. By Jhe method of images,taking oare of the presenoe of the ground, they are found to be: -ona - The Turaxwell' s equations oan also be eXl)ressed in funo.tion of the partial oapaoitiE38, in the system : Q /1 = K V + I< (V - V,) + II I t:/, I '/4. = 1, (~-";) + i\.z ""' - - ----+/<t-11. (V-V) I "'- y + • - - - + ~~ (~ - ~ ) For a three phase line, these equations correspond to the picture below. We have: K13 I<./~ ::::I<--<:/~ J' ~ The k's are positive. Case of a. tJ:;ree ;ehase line. ~he a.'s are known. Let us express the o's in fu:nctica of the a.'s. ~, i +- ~"'1~ + 4I3 1J = yI ~..:, 1t +.::'La-'!.. + tl. 13 :::Y.: -l.3 '-1...j/ 1t + a.J.t i + 4 -H 1J =Y3 ~ V.: y J 1, =- a.II I.J. a./3 q,_ ./L a.4~ d,. 4.. J4. q_.Js a.1,2,,, tt.,~ ~, ~ -l.tl, 4 a.3 I "Lj4. ct...3~ - ~, 1; +'-,.:. 1; .._,,.3 ~ ,,U and : t: II with: The other e's oan be obtained at once by permata. tion of the subsoripts. For all Jre.ct1oo.1 ]1.U'poses, it is well sufficient to take the average value of the a// o,t4. o3J : 'l!hen ; ~ =, ~ ":" + ~'(t 1- 1:j) 1~ ~ " I (r; +1:;) • '"': -90- Fore. b,ala.noed three .Phase line: Y I +Y-Z. +Y":o j I I d. and 1,:: (c -c)-r, ftI . 7 ~ 0 I., /,..,..........., .VI '- / / or J. 3 ,1. .......... .: The constant ratio o between the instantaneous 0 . . .• . oha1'ge and volta§e of any. wire ia known a.a the in.du.atrial ~8:paoitz of the wire . .1) = I or co : c - , =- ~-4Jra. +-.2~1 ( ... - 4 1 , l ( a. I +-~-<) The expression tor the ce.paoity per wire shows the influence of the presenoe_of the earth, However, in general: and OJ For a double wire line: Q // =CV o I and ..Z 4,, =c (1y _VJ 0 / ~ The capacity of the line is defined as thus: B. Self i,,ll;du~tanoes of a . syste~ of ,wires. Inductance ot two wires of radii and and of distance D be-tween their centers. By oaloulating the self induo·ta.nce vrl.res by the definition : ¢ =- .l~ outside of the , and inside of the wires by the energatio relation : ./- i;,:;•: .:r/.1t'dc- d.-c : elementary volume, internal permeability external :permeability The formula (3) corresponds tt ::the equation( ::~ ). ....Mutual induotanoe of two double lines . .• Let the two loops be (1,2) and {3,4), ha.Ying unit length. The mutual induotanoe is the flux in loop (J,4) due to the ourrent of' unit value 1n 1 and 2,. Suppose now -tha:t the wires 4 and 2 are confounded, Indi.totanee of. . a three phase balanced. line. -93- It 1a i mpossible to speak of the self induota.no~ of a three phase line in general. since there are three wires carrying three different 3 currents, Each of these wires is the origin of e,m.f 1 s of' induotion whioh may be decomposed into an e.m.f. of' se1£1nduotion and two e.m.f''e of mutual induotion. We shall see that it is possible to express the e.m.f. in one wire a.s a function of the current in that same wire, under the condition that we define the industrial self' induction of the line, We may suppose that the neuter wire exists, that it has the same radius than the other wires, and that it oooupies the geometrical center of the system. Since ~· +-'-~ -1- '-.J =-- o ou:r.rent f-'i)+(-i) +-(-";) :: o , we may su.:vpose that the flows in the neuter. The system is thus equivalent to a sum of three oirou.1ts 1-0, 2 ..0, 3-0; in which the currenta flow. < :: I (l .i:!i. + ~ di,l. -1-- ~ di-.;41': di' .Jo/to d,") di: where L is the industrial self i;nd.uote,nee of the wire. ( formula 3 ) ( formula 5 ) d~ .Jd2. whi<?h is exactly half of the value we found in (3). The formula. ( 6) oorres:Ponds to the rela:tion {1 ) • Note 5. We no-tioe that: then Using now L'Hosi,1 tal' a ru.1e : ~[/r:L: ~~1 p;;;~t.:~[~: f~,•~L: 0 In genei~al, ./~r-::J f .= o "-=-"° if_E:.a.nd.2!.>o !his means that the ei-ponentia.1 :f'unotion goes to infinity faster than any alg0braio function._ This 1s a well kno\m mathematical pro~erty, which a series expansion would also bring out. Bibliog raphy •. first Pa:i;-t. Eleotrora.agne-tio Theory ... 3 Vol. a.Heaviside Eleetrioal Papers. 2 Vol. O.Heaviside Operational Circuit Analysis. V. Bu.sh Heaviside's O:p ere.-tional Calculus Berg Philosophical Magazine,VII, p . 1.53 (1 929 ) Balth. Van Der Fol Electric Circuit Theory. Four.ier Integrals J • ~ .Carson Caml]bell and Foster. Fu.notions of a complex ~riable Ma crobert Theory of Transient Oscillations of Elec t rical ]Jetworks ( Trans.A.I. E. E. ; 8 ,1 ·91'9 ,p, 345 ) Carson .Seeond Part. ]Ja ste1~J 1 s Thesis. Ni.,I, T.June 1933 .. Theorie Der Elek-f:;rizi tt.-b Abraham u.nd Beeker ~:rans .A . I . E. E, Feb .1 919 , S·teimnetz Radiation fr om Transmission Lines {~ M. I.T.J?ublioations, c . H.Manneback Deformation of' -'Waves ( Elektro·teknik Ti dskri:t·t) Jeans J.Jle c tromagnetism AJ.):plioations Industrielles de l' Electricit& .Funkt1.o:nentafeln H.:Pleijel A. Niau.du.it E ,Jabnke und F.Emde Eleotrioal eonduotors ( p. 200 ) Perr ine. Traveling Waves Bewley