Integration Method for AC R and L 1920 needs 80 min djvu conversion

DEPARTMENT OFCOMMERCE Scientific Papers OPTHE Bureau ofStandards S.W.STRATTON. Director No.374 ANINTEGRATION METHOD OFDERIVING THEALTERNATING-CURRENT RESISTANCE ANDINDUCTANCE OFCONDUCTORS BY HARVEY L.CURTIS, Physicist Bureau ofStandards APRIL 7,1920 PRICE. 10CENTS SoldonlybytheSuperintendent ofDocuments, Government Printing Office Washington, D.C. WASHINGTON GOVERNMENT PRINTING OFFICE 1920 oy“eee BeyStet AS~=oveee%ree73q |sae, ePte=eri lea |ihe mKSStaeasPgoraeJ=awouaihoneieudhtt Peng! 2alrae»: ASSeepeewee | :ekeSo =aepoi ngapeBoieee : ie eS :ee zeeete 5tea7* DEPARTMENT OFCOMMERCE Scientific Papers OPTHE Bureau ofStandards S.W.STRATTON. Director No.374 ANINTEGRATION METHOD OFDERIVING THEALTERNATING-CURRENT RESISTANCE ANDINDUCTANCE OFCONDUCTORS BY HARVEY L.CURTIS, Physicist Bureau ofStandards APRIL 7,1920 PRICE, 10CENTS SoldonlybytheSuperintendent ofDocmnents, Government Printing Office Washington, D.C. WASHINGTON GOVERNMENT PRINTING OFFICE 1920 ANINTEGRATION METHOD OFDERIVING THE ALTERNATING-CURRENT RESISTANCE ANDINDUC- TANCE OFCONDUCTORS ByHarvey L.Curtis CONTENTS Page I.Introduction 93 II.Outline ofthemethod 94 III.Alternating-current resistance andinductance ofastraight cylindrical conductor 96 1,Derivation offormulas using realpower series 96 2.Derivation offormulas using complex power series 103 IV.Alternating-current resistance andinductance ofareturn circuit 106 V.Application offormulas toexperimental results 116 VI.Appendix: Evaluation ofintegrals anddevelopment ofseries 121 1.Toexpand log^inaFourier series 121 ^ , ,cosna . -TA• ' f^2.Todevelop ^—maFourier series of6 121 3.Evaluation oftheintegral I^ 122 4.Evaluation oftheintegral I2 123 I.INTRODUCTION Themethod heretofore used inderiving formulas forthecom- puting ofthealternating-current resistance andinductance of conductors requires thedetermination ofthedifferential equations ofthemagnetic fieldandthesolution ofthese equations imder the conditions imposed bytheshape oftheconductor. Formulas havebeen derived foronlyalimited number offorms.^ Themethod outlined inthispaper isoneofintegration. It requires thattheconductor bedivided into infinitesimal filaments bysurfaces which coincide with thelines ofcurrent flow. The magnetic field atanypoint isthesum ofthemagnetic fields ofall ofthese filaments. Thecounter-electromotive force inafilament isdetermined bytherateatwhich themagnetic fields ofallthe other filaments cutthisfilament. *Themost important ofthese havebeen collected byRosaandGrover, B.S.Bulletin, 8,p.17a;1911; Scientific Paper No. 169. 93 94Scientific Papers oftheBureau ofStandards [Voi. i6 II.OUTLINE OFTHEMETHOD Ifsmall wires areplaced sidebysideandanalternating electro- motive force applied totheir terminals, thecurrent which flows in anyoneofthewires willbedetermined notonlybyitsresistance and inductance, but alsobythecounteivelectromotive force caused bythemagnetic field oftheothers cutting this wire. If there areonlytwoidentical wires, thecurrent willbethesame in each. However, ifthere arethree ormore, theymaybeso arranged thatthecounter-electromotive force insome isdifferent from that inothers, thereby causing thecurrent tobedifferent in different wires. Iftheresistance and self-inductance ofeachwireareknown and themutual inductance between thewires taken inpairs isknown, equations insufficient number canbesetuptosolve forthectirrent andthephase ofthecurrent ineach wire. Forexample, ifthere arethree conductors whose resistances are r^, r^,and fj,whose self-inductances are Z^, l^,and /g,whose mutual inductances are Wi2, ^23* aiid ^i3»^^^whose instantaneous currents areI\,I\, and 7^3: ^, r, idl\ dl\ dl\ '=^'''^+^^t/+"^-7^ +^-"^' dr, di\ di\ dJ-^'^^^-dl-^'^^^-dJI\n+k-^~+W13-T-' +m,. where Tosolve, assume thatE'=Ecos oit. /'i=/i cos {(Jit~-<i>^) I'2=12cos (Cx)t—(f>2) r^=h cos (o)t -</)j). Substituting these values intheequation, there result three TTequations when w/=oand three independent oneswhen oit-=-i making sixequations fromwhich todetermine thesixunknown quantities; viz,themagnitude andphase ofthecurrent ineach of thethree wires. Asolidconductor ofinfinite lengthmaybeconsidered asmade upofaninfinite number offilaments, andeach ofthese behaves as though itwere aninfinitesimal wire carrying acurrent. The Curtis] A.C.Resistance andInductance 95 instantaneous cunent inanyonefilament isdetermined from the equation E^=r8j\+lo~^br^ +jJM.y |5/V,(i) where E^istheinstantaneous electromotive force, rand /othe resistance and self-inductance ofthefilament, 5/'xthecurrent in thefilament, M^y themutual inductance between thisfilament andanother filament atYinwhich thecurrent is8I\. Ifthere areother conductois inthefield, theintegration must include these also. Having determined 5I\,thetotal current /'through the conductor isthesumofthose through thefilaments, or '-n'''^-n'''^'^(2) where f/'x isthecurrent density atxanddSanelement ofthe cross section oftheconductor. Inequation (i),several ofthequantities approach either zero orinfinity asthearea oftheelement approaches zero. Hence it isnecessary toexamine eachterm oftheequation andtoretain onlythose which haveafinite value. Intheterm r81\,r=-j^and 8I\=U\dS,where aistheresist- ivity ofthematerial, /thelength anddSthearea ofthefilament, andU\thecurrent density. Hence r8I'^^U\ <r/,which shows that thisterm isaconstant andindependent ofthearea ofthe filament. Theterm /«-j.8I\isthecounter electromotive force caused by thecutting ofthefilament bythemagnetic fieldwhich thecurrent 5/'xsetsuparound andinthefilament. Ifthefilament becomes infinitesimal, thecurrent, andhence themagnetic field, isinfini- tesimal, sothat inthelimit thistermbecomes zero. filament atXcaused bythecutting ofthisfilament bythe magnetic fieldwhich issetupbythecurrents inalltheother filaments. This isafinite quantity, although M^jbecomes infinite whenYapproaches X. Ifanalyzed mathematically, it isfound that this isacasewhere theintegral ofafunction which hasoneinfinite point isafinite quantity.Theterm| |Mxy -j.81'jisthecounter electromotive force inthe 96 Scientific Papers oftheBureau ofStandards [Voi. i6 Itfollows that equation (i)may, bysubstituting for5/'y, U'ydSy bewritten intheform dU\E^^U^^al +jjM.y'^^dS.(3) Thisequation maybeused todetermine thevalue ofZ7'xincon- ductors ofknown form, provided thenecessary integrations can beperformed. Having determined L'^'x, equation (2)—viz, /'= I IZ7's c^S—gives thetotal current intheconductor. Equation (3)applies only toconductors which aresolongthat theend effects maybeneglected. Toapply (3)toaparticular case, itisnecessary tobeabletoexpress M^yasafunction ofthe distance between thefilaments, and thiscanbedone onlywhen thepermeability oftheconductor isunity. III.ALTERNATING-CURRENT RESISTANCE AND INDUCT- ANCE OFASTRAIGHT CYLINDRICAL CONDUCTOR Formulas forthealternating-current resistance andinductance ofastraight cylindrical conductor ofinfinite length have been developed byseveral investigators.^ In allcases they have started from thedifferential equation ofthemagnetic field. Themethod ofintegration outlined above hasbeen applied to thiscase todetermine whether itwillreadily give useful results. After thecomplete derivation using realpower series hadbeen completed, itwasfound thatthework might bemuch simplified bytheuseofcomplex power series. However, thecomparison ofthetwomethods isamatter ofsome interest, sothatboth are given. 1.DERIVATION OFFORMULAS USING REALPOWER SERIES LetPxandPybeanytwopoints inthecircular cross section of aconductor oflength /,and letPx0^andpyBybethepolar coordi- nates ofthese points.^^^^^ Fig. I.— Thecross section ofacylindrical conductor, showing coordinates *Thedifferent formulas andthemethods ofreducing fromonetoanother aregiven inthepaper byRosa andGrover akeady referred to.* curits] A.C.Resistance andInductance 97 Ifdisthedistance between P^andPy, d=VPx'+Py'-2PxPy COS (Oj,-dj); also,^ =2/(log 2I—1)—Ilog c^^ =2I(log 2/-1)-/log[px'+p/-2p^py cos(e^-dy)] Assume that E^=Ecos cot, .'.U\=t/xcos (cot—<f)x) and Z7'y=Z7ycos {0)t—(^y) Substituting these values andthevalue forM^y inequation(3), thefollowing equation isobtained: Ecoscot =<rl[7xcos (oj^—0x)+ IPyC^Py IUl(log 2I—1) —Ilogpx^+Py^-2pxPy cos(^x-^y)[|—o)Uy sin {cot—(f)y)\ddy =<tIZ7xcos0xcos cot+clU^sin^^sinat +O)IPydpy IIII(log 2/-l)-/log Px'+Py^-2PxPy COS ((9x-^y) Uysin<^yCOS oot—Z7yCOS <^ysinco^c^^y.(4) Thisequation determines thevalue of^xand <^x. Tosolve theequation, assume that^xsin^xand Z7xcos <^x (andhence corresponding values ofUyand <^y)canbedeveloped inapower series withundetermined coefficients. Bysubstituting this series inequation (4)thevalue ofthecoefficients canbe determined. Since thevalues ofUand 4>aresymmetrical around thecenter, Bdoesnotenter intothevalue ofeither, hence theassumed power seriesmaybewritten Z7cos=ao+bop+CoP^+dop^+Cop^+/oP^+gop^+Kp'^+iop^+ L^sin0=Ao+^oP+Cop'+I^oP'+£:oP^+FoP^+GoP«+//oP'+/oPH Substituting these values in(4),eachterm iseither anintegral containing onlysome power ofpyoranintegral ofthetype I^, given intheappendix. Making theintegrations andputting 2/(log 2/—I—log a)=Loo{^^^inductance duetothefieldoutside •B.S.Bulletin, 8,p.151(Scientific Papers No.169). 98 Scientific Papers oftheBureau ofStandards [Voi. 16 theconductor, which istheinductance atinfinite frequency;, equation (4)becomes Ecos oit=alcoso)t{ao+Z>oPz+CoPx^+ } +alsino}t{Ao-\-BoPx +Copx+ } _ /Aoa' ,Boa' ,Coa' .\+27rcoLoo coscon11 1-j —27rwLoo smco/i1 1 1- ) \2 3 4 / . /Aoa' Boa' Coa* \+47rcol coscon11^+)V4 9 16 / 7JAoP.' BoP/ CoP.' \—Airm cos cot I1 1^-+ I V4 9 16 / . /aoa^,60^' ,^oaV \ V4 9 16 / +4^^^ sin^^(^-^ +—+-75-+y(5) Ifost««=o,COS CO/=I,andsin co/=o,then putting rj=— ^=ao+6oPx+coPx^+ •••+—r-V"!"-^T^-T"^ ) ^/Aoa'Boa' Coa* \+41711-+—7-4- •••I \4 9 16 / TT IfCO/=-»COS CO/=oandsin co/=i 2 o=Ao+5opx+Copx^+ ^—(-^~r"^~4~—/ /aoa^ ,^oa^,Coa* . \ V4 9 16 / +4.(^+^V-f +....) (7) The coefficients oflikepowers ofPxmaybeequated inboth (6)and (7).Theequations thusformed aresufficient forthe determination ofthevalues oiAo,Bo,Co . . .andao, KyCo . • . . Since bothBoand boarezero, allcoefficients ofoddterms ofp, arezero. Theremaining values aregiven inTable i. curtts] A,C.Resistance andInductance 99 TABLE l.^<}oefficients ofPowers ofp^Taken fromEquations (6)and (7) Power ofp Coefficient from (6) Coefficient from (7) Px"^r^-'-^i^-^-f-•)„^27,Loo/aoas. Coa<. 0=Ao^(^2+4-^••) ,..,(^%^V.. )/aoa2 Coa< ^ ^\4„^4^16t• • •; P.« Co=AoJ7 Co=-ao77 Px<„ Co»7Ao *?'Eo4-4 Px« 8°'"9 36Co 77aoij»^°9=36 Px8 ^16 576Jgol7„A„„* 16 576 p,to x„=( 1)°^"^^"^'^ °^ ^22.42.63--(to)!V ,,^n Ao(4»;)2»^-C1)22.4^.6»..(4n)» Px^+2 '^ >22.42-62--(4n+2)« Note.—xoandyoareused, respectively, asthe^nandthe4W+2 letters ofthealphabet. Substituting thevalues ofthecoefficients obtained from the higher powers ofPxintheequations forthecoefficients ofPx°,the following equations result: (tIao+217L00 /[AoO^ aooa'^ Aoffa^ aori^a^ ~~2 4247""^288"^ +(-i)n4^ Aoff'' a^'^^^ (-1)'2n+l /i4n+4aoV a 4^+2 2^•4^-6^ ••(4^)2 4W+4 22-42-6^- •(4n+2)^ ^14 16 144 2304 4'^ Aq ??^^a%-^^] o=Ao-(4W+2)22^•4^•6^ ••(47i)• i2n+l 2n+l /,4n+4ao??"""* a (41^+4)22=^.4'. 6'-.(4n+2) 27?Loo+] (8) /tapa^ApTf a* apYj^a" 2 4 24 +(-i)n4^ aor;2^a^^+2 +(-!)'^n+222-42 •62-•(4^)2 4^+422-42•62- •(4ti+2)' tttoa2Ao 17a'^aprf a"+] +(-i)^ap772^a4n+2 (4^+2)222.42.62. -(4^)2 (9) 148921°— 2( loo Scientific Papers oftheBureau ofStandards [Voi.x6 Inorder tosimplify these equations, thefollowing substitutions niaybemade: ^72^ ^'^ ^^n+i22.42-62...(4^1+2)2^••••^^^ ^144^ ^^ ^^(n+i)'22-4'-62-. -(4^+2)2^••••' i-j2"^'**"^^!>>(2n+i)22-4=^-62.-- (47^)2^•••• ""^ Then equations (8)and (9)become Solving foraoand >lo ''^^Vin'a'LJV, 7,'a'X, £ 2/ _9r "\7" Since thecurrent, 5Z^',through thefilament atP^is Z7x'PxC^Pxfl^^x, thetotal current /'through theconductor atanyinstant isgiven bytheintegral ^'-{^^^-'={ ^^- Pxc/Pxde^' =IPxc?px I (ao+CoPx^H-••••)cos co^ 4-(Ao+CoPi'+)sinojMJ^x Integrating T, ^ / X /aoa^ Coa* \/'=/cos (co^-</)) =2x(-^+-^— + jcos w^ +27r (++jsm w^ (12) Curtis] A.C.Resistance andInductance loi Hence putting successively 03t=oandoot=- /cos0 =27r(-y+—+. . .J=27r(—Y,+-~~W,) (13) /sm,^ =a.(4f +^V...) =..(^V,-M?!w'.)(14) P„,,.j(^V^V.. .J+(^+C^V .. .. Jj =4tM«o^+^o^)(^V,^+^W-,^^ (15) Inanalternating-current circuit theeffective resistance Ris determined bytheenergy loss,or PR=E Icos<f) .„E/COS0 (16) ••^=p Likewise theinductance isdefined bytheequation .EIsin(f)^L= j^(17) Substituting values from (13), (14),and (15)in(16)and (17) E(a.Y,^^w) R= ^7^\ (18) e(a.y,-^-^w) coL=^ ^ y-i^^-T (19) ira'(ao'+A,') Iy^+^-^ W,') Substituting thevalues ofAoand ao al 2/' '42/' '2 ^ 'K=--—— o4TT7 2 (20) '4 ^al I^' * '2 ^ 4/ ^ 8 , .ajL=—» ,.xx/o (21) '^4 ^=^^o— .2.4M/> (22) 1^1^+ I02 Scientific Papers oftheBureau ofStandards {Voi.i6 Where Ro,representing thedirect current resistance, equals—- Tza '^4 Where L^,representing theinductance atinfinite frequency, equals 2/(log 2I—1—log a)and coisgivenbytheequation err;—=co. TT Substituting thevalues oiW^, X^,Y^,andZ^ I+V-+V+-5^—+ r)_o6480 181440 1+-^— +-^^ +^^+121440 725760 ,I+-^+-^ 4-—^ +r_r/ 244320 2903040^2 Ty^a^ r;^a« ty^a^^(25; 121440 725760 These correspond exactly withtheasymptotic formulas ofRussell.* Tables tofacilitate thecomputation have been published by Savidge^andbyRosaandGrover.® The series W^,X^,Y^,andZjmaybeexpressed interms of theberand beifunctions and their derivatives. Byputting q^=^ria^,andcomparing them termbyterm, thefollowing re- lationships hold: W,=-^h^r^q (26) X,=|^(i-ber^) (27) Y,=|bei'g (28) Z,=^heiq (29) Substituting these inequations (22)and (23), thewell-known solution interms oftheberandbeifunctions result.^ *Phil.Mag., 17,p.534; 1909. »Phil.Mag.. 19,p.49;1910. •B.S.Bulletin, 8.pp.173-136 (Scientific Paper 169). 'B.S.Bulletin, 8,p.175. Curtis] A.C.Resistance andInductance 103 Therefore, foracircular conductor, themethod ofintegra- tion gives results identical with those obtained byprevious investigators. 2.DERIVATION OFFORMULAS USING COMPLEX POWER SERIES Thederivation oftheformulas maybemuch simplified bythe useofcomplex quantities. Let i«t ,,, J.Ji(«t-0) dU^ .JJi{u,t-4>) Where Substituting in(3),viz: E Et'=<7iuJ^\-'^ f(M.yic,uJ'''r'^dS Take out e'"andinsert thevalue of Mxy=2/Hog^-lj =2l(log2l-i)-l log d'^ E=<TlUy,e'^+ta) Ipjdpj I12/(log 2/—i)(30) -/log[p,2+p^2_2p^p^ COS{e^-e^u,e "^de^ (31) Assume ^7,€"'^^ =A+5p,+Cp,2+L>p,3+.... Where A,B,C,D,,etc., arecomplex numbers. Substituting in(31)andputting E=<jI(A+Bp^+Cp^'+Dp^'+) ,.,rAa',So' Ca' "I +4,,,/|^_+_ +_+ ....J -K¥-^^^— •](3.) I04 Scientific Papers oftheBureau ofStandards When Pi=o ^ ,,.,VAa-" Ba' Ca' 1 "L23 4 J .TAa' Ba' Ca'"1iVoL i6 (33) TTCOEquating theHkepowers ofPxand letting—=77,thevalues ofallthecoefficients maybeexpressed interms ofAasindicated inTable 2: TABLE 2.—^Values ofCoefficients ofPowers ofp^inTerms ofAfromEquation (32) Powers ofp Coefficients P. B=0 P«» a/C=AIa>WcJ'-'^-,,A P.« D=0 P><./E-^*;-'a.»^iA v*A P>> F=-0 Px«WG-^'r'la,3x3A_ I,»A^ 4-9<rJ (3I)« Pk' H-0 Px" _GlcoW'^•'16a,«^4A ^4A 4.9-16a< (4I)> Px^-1 Q-0 Px2" l-„-A Note.—QandRareused, respectively, asthean-iandanletters ofthealphabet. Substituting these values in(33): *L2 424288 (?z!)2(2n +2)J .,.ra\ir)a' rj^a^ iri^a\,^'^r7^a2^+2-| ^ |_4 161442304 [n!(2n+2)P J^'^^ Todetermine thetotal current through theconductor = I\^dp^\e'-KA +Cp^'+Ep^' +--')de^ Jo Jo Integrating: rat^^ :y^^' Ca'Ea' \ V2 4 6 / Curtis] A.C.Resistance andInductance 105 Simplifying andsubstituting values ofC,£",etc.,fromTable 2: ri^ a/"^^ ,^Va^ V^^^ ^V^^^ ,^^^^^ .\ / ^/e-.*=2x^(^- +—-—-^+^^+--j (35) E' EAlso~jr=R +io)L,or~-z:r^=R +iccL (36) Letting Pl^i,Xi,Yi,andZ^represent thesame values asbefore, then substituting from equations (34)and (35) inequation (36): irja^Wi Rationalizing thedenominator bymultiplying byY^ andtaking therealparts .^ , ^na'X.Y, . , ,naW,Z, R ^-4——^ ? (38) Letting—2==-^o ^^rl^W^_rMX^ R=^- ^,,„., ^ (40) Inlikemanner ^=-L» +-TTw^^(41) 2 5WV '"*" 4 These equations areidentical with (22)and (23)showing that theresult using complex power series isthesame aswith real power series. io6 Scientific Papers oftheBureau ofStandards Wol.x6 IV.ALTERNATING-CURRENT RESISTANCE AND ANCE OFARETURN CIRCUITINDUCT- Ifareturn circuit consists oftwoparallel cylindrical conductors, whose length isgreatcompared tothediameters ofthewiresand tothedistance between them, themethod ofintegration canbe applied tothedetermination ofthealternating current resistance andinductance ofthe circuit.^ Ifthewires have thesame diameter, thecurrent distribution inonewire issymmetrical about thelinejoining thecenters ofthetwowires. Iftheequation ofcurrent distribution isgiven inpolar coordinates, itwillbe identical forthetwowires iftheangles aremeasured from theline joining thecenters ofthewires. Fig. 2.— Thecross section ofareturn circuit, showing coordinates LetPwith coordinates p,^beapoint inoneconductor atwhich thecurrent density istobedetermined, P^another point inthe same conductor, andP2^point inthereturn conductor whose center isatadistance sfrom thecenter ofthe firstconductor. LetU\U[,andU^designate theinstantaneous current density and Z7,U^,andU2themaximum current density atP,Pi,andPj, respectively. Also letm^represent themutual inductance between thetwofilaments atPandPj,whose distance apart isd^,andm^ thatbetween filaments atPandP2,whose distance apart isd^. Equation (3)forthiscasebecomes E'=U'cl+jjfn,^" dS,-fjm,^'dS, (42) where dS^anddS2areelements ofarea atPiandPg. Themutual inductance between twolongfilaments•is m=2/ (log 2/—i)/logd' ^Nicholson haspublished aformula covering thiscase (Phil. Mag., 18,p.417; 1909). However, the author hasnotfound anyrecord ofitsappUcation toexperimental results, and efforts tocompute byit indicate thatthecorrection factor, duetotheproximity oftheconductors, isnotonlymuch toosmall, buthasthewrong sign. »B.S.Bulletin, 8,p.151(Scientific Paper No. 169). Curtis] A.C.Resistance andInductance 107 Substituting values ofm^andm^in(42) E'=U'Gl+2l(log2/- 1)rr^^'dS,-l{ flog^1'^''^5i -2/ (log2/-1)JJ^^'JS^+zJJlogc^,^^^' ^5, (43) Since thecurrent inthetwoconductors isthesame, ^{u\ds,= {{u'.ds,. Sothatterms twoandfour ofthesecond member of(43)cancel. ButE'=Ee-\ U'=^e^C^t-.^), Ui=^^e^^"*-^^), etc. ^i'=P^+Pi'-2pPiCOS (^-^1) d^={s—pCOS^—p2Cos^2)^ +(psin^—P3sin B^^ =9'+P2'-2gP2cos{a-e^) , where qcosa=s—pcos6 andqsina=psin^ sothatq^=s^—2spcos^+p^ Hence equation (43)becomes E=^(tIU€-''^ f*a. r2T -ioill p,dp,\ log[p^+p,^-2pp^ cos(6-6,)] Ui€-''^'dei Jo Jo +ioil Ip^dp^\log[q^+P2 -2gP2cos(«-6^)] U^e'^'^'dd^ (44)Jo Jo Both L^and arefunctions notonly ofpbutalsoof6.Butas L^and </>have thesame values for—^asfor 6,thefunctions of 6when expressed inaFourier series willbeinterms ofthecosine series only. Hence, assume ^€-i<^=ao+«!cos6+a2cos26+ +p(6o+^1cos^+62cos2^+• • ••) +P2(co+Cicos6+€2cos 2(9+• •••) ^^"^^ +••( ) Ifthese values aresubstituted in(44)andtheintegration per- formed, acomparison ofcoefficients shows thatmany vanish. Equation (45)reduces tothefollowing: Ue-'"^=ao+CoP'+eoP'+goP'+ +(6iP+c^iP'+/iP'+••)cos^ +(c2P^+e^p^+g^y+••)cos28 +(d,p'+f,p'+h,p'+--) cos3(? +(e,p'+g,p'+iy+••)cos 4(9 +()(46) 148921°— 20 3 io8 Scientific Papers oftheBureau ofStandards [Voi.16 Substituting (46) in(44)andintegrating byusing theintegrals given intheAppendix, E=(Tl{ao+CoP^+Cop^+QoP^+ +[b,p+d,p^+f,p^ +h,p'+ ]cos^ +[C2P^+e^p"^+g^p^+^2p^+••••] cos20 +[d,p'+fsP'+h,p'+;>«+• •]cos 3(9 +} f /aoa^ Coa* ,Coa^ Qoa^ \[-^c^l\4^logai^— +—+~^+—+• • • • J| .AVb.p^ d,p^f,p'h.p'' x^p'^^^' l rL8 24 48 80 {2m-\-iy-iJ I |_12 32 60 96 /\.nr—^J -7rp2-^+^-+^+-V+--•+-^^ +•••• cos 2(9 |_2 4 6 8 2W-2J) ~zco/4x-^4-'-^ +-^+-i5^+.. .+-^^^— +• •• [ [_1640 72 112 (2m—1)2—9 J Z [_2 4 6 8 2m-4 J^ —tco/jterms incos^6,cos 5^,etc.[ +zc./J4^logq[^—+-_+^+-y+... .J .A2irrc2a^ e.a^ q.a^^ i.a^^"1cos 2q;1 (2[_6 8 10 12 J^ ) 3L8 10 12 14 Jcos 30: ico/jterms incos—^»cos—j,etc.[ (47) Curtis] A.C.Resistance andInductance 109 Byequating tozerothecoefficients ofliketerms ofpand^,many oftheundetermined coefficients areeasily evaluated. Themost important ofthese aregiven inTable 3: TABLE 3.—Coefficients ofTerms Involving pand 6fromEquation (47) TTCO Let— =v (T Coefficients of— Values obtained Coefficients of— Values obtained P» Co= i7;ao p^cos 2d„ii7C262=-3- P* IrjCo TJ^aoeo=-4—— p^cos 26 ^-8 24 "' ^ it/eo i7??3o«•"9 35 pScos 29.i7?g2 i77^C2 ^^15""360 ffi *"16 576 piocos 29 ^»-2?-8640 plOiTjio i77=ao *^*'°"2514400 pl2C0S 29 _ i7jk2 i775c2"^'^35^302400pU ivyko 7?^ao"^o"36~518400 places 28 48 14515200 p»COS<?d,-^''^^^2 P^cos 3^b-'f p'COSff ^'6 12 P'cos 35^-'-^'-'^ picas B^^-12 144 p9C0S 35 '^18 720 pScos e iTjhi Tj^bi "20°"2880 piicos 35,ivk V*d3li=—^-^=——- ^2820160pllCOS^ *^3086400 piscos 35 i^l3 irjSds^40806400 piscos e i»?li ^«bi °^42 3628800 piscos 35 l77n3 r;«d3 ^^54 43545600 Substituting thevalues fromTable 3,-theseries inequation(47) containing powers oftheradius, a,reduce asfollows: 2468 a,a\coa* e,a\g,a\ 4 16 36 64af,a' 2 12 ~744"^2880 86400"'' aoaT trya^ rj^a* I+-— 36 t77^a^ ?7%^ ^T7^a^^+-^^+—4^+ 576 14400 518400 2'^4"^6'^8"^"" 2L4t7]a^ rj^a^ 36 ir]^a^ Tj^a^ irj^a^^ 576 14400 518400 no Scientific Papers oftheBureau ofStandards c^ e^a^g^ ijP^ ^c^a^V irja^ rfO*" 2 4 6 8~2L6 72 irfa^+77%^+ir^V^ 1440 43200 I514400 24.68 2Lii7a^ 77^a* ^r;^a^ ry'^a^+ +120 61a* d^a^ f^a^ h^a^^ 4 6 8 102880 100800 4838400 ^—1 I+- 3 24 f[ irfa^ ri^a^ irj^a^^ 360 8640 302400 6 8 10 126L440 720 20160 806400+ dsa\isa'\ha'\ua'' d,a' — \-- 1 p. . . .=—— 8 10 12 14 8I+irja^ 77-a* S60 irj^a^ 7]^a^ irj^a^^ 1260 40320 I814400[Vol. 16 _c^a'=2^0 -'-fD^ ^h,a^ =6^^ .¥^. Itwillbenoted thattheseries areallarranged sothatthe first term isunity. Itcanbeshown thatthey areallconverging series. Substitute theseries given above in(47)andretain only the constant terms andthose oftheform p^cosnd,since thecoefficients ofallother terms arezero. Also insert expansions oflog q,of cosa.cos2a• •^t, a- t^ a.'^^^u »of ^— >etc., asgivenmtheappendix. Put -q=—.inen al=aQ+b^pcos6+C2P^ cos2d+d^p^cos3^+• ••• —irj[2aQa^AQ loga—aQa^A^—b^a^A^p cos6—c^a^C^p^ cos26 -—d^a'D,p^ cos3(9+- •] o +ir][2aoa'A, (logs- jcos^-^^ cos2^-^3cos3^)] -ir]\——^2 3"1 (l+—COS e+~cos2^+^COS 3^+• ••) S o O_J 2p 3P 4P^ (l+^COS^+^COS20+-^ COS3^+••)] 3P 6p Iop'(1+-^-cos6'4--V cos2^+-f-cos 3^+-••) (48)] curtts] A.C.Resistance andInductance iii Thetotal current through theconductor isthesum ofthe ciurents through thefilaments, or r=^^hr=^^u'pdpde =e^-'l pdpl {a,+c,p'+e,p'+--'- Jo Jo +{b,p+d,p'+f,p'+'-) cos (9 +(terms incos20,cos3^,etc.)}c^^ since theintegrals ofthecosine terms areeachequal tozero. But .'.Ie-''^=Tra'a^Ao (49) Thealternating-current resistance andinductance ofacircuit isgivenbytheformula j^,=R+io:L (50) Bysubstituting (48)and (49) in(50)andequating tozerothe coefficients ofeachpower ofp,thefollowing equations result: R+uaL=— 2—-r-\(io+2ir]a^aQA log—+irja^a^AjTraaQ/\ot ^ ,.(c^a'C^ a^a^A^ h-fl^B^ c^a'C^ d^a^Dj \o=c,+tn{^—-J,—, —, —, j(53) ./d^a'Do agpa'Ao b^a^Bi ac^aTi S'^so'D, \o=d,+tv\^— —,^^5~,^, j(54) Inorder toevaluate RandL,itisnecessary toeliminate a©, 61,C2,anddsfrom equation (51). Toaccomplish this,solve fordg inequation (54)andsubstitute intheother three. Then solve for C2inthethird oftheresulting equations andsubstitute in 112 Scientific Papers oftheBureau ofStandards {Voi i6 theother two. InthiswaytheeUmination maybeaccomplished. This isouthned below. Solving (54)ford^ ./2aoaMo .b^a^B^ 2c^a^\ /2a^^Ao b^a^B^ 2C2a^CA Bi-ds 17] 3iva^Di(55) 6s' Substituting thisvalue ofd^in(51), (52),and (53)andrepre- senting theresistance todirect current byRq=—^»equations (57),7ra (58),and (59) result. R-hicoL I I • , ^ , <y ., >. —-^—=X~a\^'"^^'^va'aoA log-+tva'^a^, -irja^ rb.B^ a^c^C^ i'(]a^DJ2a^^ b^a^BX V[ , x sL2 66-"*I2y» \2,s^D^'^ 2s'Dj]\^^^^ o^b.^i\a^b.A.-'-^-'^^^ (58) o-r ,,• (^o^'-^o b.a'B, c,a'Co c.a'CA Solving (59) for c. ( ,,b,a'B\ \2 26-^/^2="^—Tr -^ir^ (60) 1+i-na^ \^ ir}a^Ci ^1 ^r;aY.,b^a^BA ., .then c,=;j:^^(^^oA +^^)(61) Curtis] A.C.Resistance andInductance Substituting thevalue ofc^inequations (57)and (58) =-7-+-^—+2ir)a^ log- R. A, Ao 'a irja* (i-qaQa^A^D^ i'qh^a^B^D^ sa.A, i8^^D. 24J-^Dj ^i-^iAtW^I^A ,a^Cy irja^a ^2y'^4s'Dj'^ 6s\s'C o=bi+ir]bia^Ai —i'qh^a^B^ ^irjaoa^A2S irja'^Ci /irjaQa^A qirjbia'^BiX r)^a(,a^ ^D ^. 3S'V^'C^"^2S^C:,)^6?D~ Solving (63)for h^, 2ir]a(,a^AQ 'r)^a^,a^A^yC^ r/^aoa^MoZ^i biSs'C, 6s'D, ,,<,,,,„!^,5|^, T 4-o 1 ,•2Z1iva'B.n'a'^B.C,Let^2=1+tr?aM 1^+ 2i"' then ^1=Ss^C^ 2ir]a^a^AQ Ty^aoaMoCj rj^aQa^^AoDi sB, Ss'B^C^ es'B^D, substituting thevalue of61inequation (62) R-\-io:L l+irja^A, .,, s-^— =—^^^+2zr;aMog- ir}a^{''2i7]a'^ rfa^C^ yfa^^D^^ 5i irja^B^Ci irja^B^Di irja^B^D^+ I2i"*C,+ Ss'D..+ 24,^' 6s'C, ^iSs'D, irja^A.. ,,s—.+2it]a' log- /in Cl irja'^liTja^D^i-qa'^B^ rj^a^BjCi irja^C^] 7~\i8s'D,'^ B,~3s'B,C,'^'6?C~2\113 (62) (63) (64) (65) (66) (67) (68) 114 Scientific Papers oftheBureau ofStandards [Voi. i6 Let rya^=X thenequation (68)becomes •Where 212144 2880 86400^'^ Ai=l+ 72+ +o+ (71) 436576 14400 518400 „ ,iX X2 tX^ ,X^ ,iX^-, 324360 8640 302400^' ^ c..i+!^-^-i=^+-^ +-4^+(73)440720 20160 806400 ^ ^ i\V i\' X* i\' ,, 5601260 40320 I814400 „ ,.^,iKB^(a\ \^B^CJa\ ,, 2 ^-^'-^'(7)' (77) Separating thepreceding series intorealandimaginary parts, and representing therealpartbythecorresponding capital Greek letters, andtheimaginary partbythecorresponding lower case Greek letters as: Ao=Ao+a:o'Z'' Ci=ri+7it A1=Ai+Qiit C2=Ta+72^* ^2=B2+i^a^' 1^2=^2+h^i then therealpart ofequation (69)gives theresistance atafre- quency of27ra;, as: /aYB,B2+^,/ ^^\S) B2^+^2^i?Ao-X(AoQ;i-A,q!o) /a\B,B2+i3A 4xYaVrir2+7iT2 6V^yr2^+72= XYaY (B2T2+fe 3V/ (B2T2+iS2r2)^ +(B2r2-/32T2) 18V/ A_^.- ,.(B2T2+^2r2)(B,r,-iS,7i)-(B,7i +^:r,)(B2r2-/52T2) --^'^'%^U2A--+^A(y8)2+52 Curtis] SinceA.C.Resistance andInductance coL__Lircoa^_Lrja^LX115 al I I theimaginary part of(69)gives theinductance atafrequency of 27rco, as AoA L=2/log-+/a.+ao(«i--|) Ao^+OJo'+/Xifi-iB2-BA /X/aVvi '^6\sri ^l\Ya\ \B,T,-^,y,){B,} 3\V(B272T,-p,7,)+(B,7i+g,r,) (B372+ftr^) (79) Itshould benoted thatthevalues ofRandLasgiven inequa- tions (78)and (79)areforoneoftheconductors ofareturn circuit. Thetotal resistance andinductance aretwice these values. Although equations (78)and (79)givethealternating current resistance andinductance ofaretimi circuit atanyspacing ofthe wires, yettheseries involved arenotconvergent forhighfrequency with large wires very close together. Thefollowing table gives thehighest frequency forwhich the series areconvergent when thewires areasclose together as possible: TABLE 4.—Highest Frequency atWhich Formula WillHold forDifferent Diameters ofWire asClose Together asPossible A.W.G. No.Diameter inmilsDiameter in centimetersHighest frequencyA.W.G. No.Diameter inmilsDiameter in centimetersHighest frequency 0000 460 1.168 1000 18 40 0.102 110000 000 410 1.04 1300 .20 32 .0812 200000 00 365 .927 1600 22 25.3 .0642 340000 325 .825 2000 24 20.1 .0510 520000 2 258 .655 3400 26 15.9 .0404 830000 4 204 .518 5100 28 12.6 .0320 1300000 6 162 .411 8000 30 10.0 .0254 2100000 8 128 .325 13000 32 8.0 .0203 3300000 10 102 .259 20000 34 6.3 .0160 5300000 12 81 .206 32000 36 5.0 .0127 8500000 14 64 .162 52000 38 4.0 .0101 13000000 16 51 .129 82000 140 3.1 .00786 22000000 ii6 Scientific Papers oftheBureau ofStandards [Vcl. i6 V.APPLICATION OFFORMULAS TOEXPERIMENTAL RESULTS Thealternating current resistance ofareturn circuit ascomputed byformula (78)hasbeencompared withtheexperimental results which were obtained byKennelly, Laws, and Pierce.^<^This comparison isgiven inTable 5.Thetwovalues agree within the experimental error, except at1000 cycles with thewires close together. While the series areconvergent atthese values of spacing andfrequency, yet itwillbenecessary todevelop alarge number ofterms toobtain anaccurate result. TABLE 5.—Resistance ofaNo.0000 Solid Copper Wire asMeasured byKennelly, Laws, andPierce andasCalculated byFormula (78) Spacing 0.03cm 0.8cm 6.4cm 20cm 60cm [Calculated 1.019 1.017 -.2 1.572 1.590 +1.1 2.908 2.688 -8.21.010 1.012 +.2 1.307 1.295 -.6 1.982 1.928 -2.81.005 1.008 +.3 1.183 1.184 +.08 1.680 1.700 +1.11.005 1.006 +.1 1.176 1.180 +.3 1.662 1.690 +1.71.005 60cycles Measured 1.004 Percentdifference —.1 fCalculated 1.175 400cycles 1.175 [Percentdifference (Calculated 1.660 1000cycles 1.670 Percentdifference +.6 Measurements oftheinductance ^,nd resistance ofaNo. 2 copper wire intheform ofareturn circuit havebeenmade inthe inductance andcapacity laboratory oftheBureau ofStandards byC.N.Hickman andMiss C.Matilda Sparks. Measurements weremade atfour frequencies foreach ofseven spacings. The measured andcomputed values aregiven inTables 6and 7.They showanagreement within experimental error atallspacings with thepossible exception oftheinductance at3000 cycles with the wires very close together. This isprobably caused bytheslow convergence oftheseries. 10Trans. A.I.E.E.,34. Part II,p.1953. Curtis] A.C.Resistance andInductance 117 TABLE 6.—^Ratio ofAlternating-Current Resistance toDirect-Current Resistance of aReturn Circuit [Length ofcircuit= 1716.3 cm.No.2Cu.wire. Diam., 0.651cm] Spacing between FrequencyR/R» measuredR/Ro^computedPercent differenceValue ofeachterm inequation (78) wires in centi- metersFirst termSecond termThird termFourth termFifth term 500 1000 2000 3000 500 1000 2000 3000 500 1000 2000 I 3000 500 1000 2000 3000 500 1000 2000 3000 500 1000 2000 30001.116 1.350 1.883 2.403 1.083 1.255 1.740 2.111 1.050 1.172 1.472 1.789 1.034 1.143 1.402 1.705 1.032 1.133 1.383 1.635 1.032 1.120 1.361 1.6191.106 1.350 1.898 2.390 1.082 1.274 1.715 2.114 1.050 1.174 1.489 1.792 1.039 1.141 1.416 1.692 1.034 1.124 1.379 1.641 1.031 1.115 1.362 1.6180.9 .3 .5 .1 1.5 1.4 .2 1.1 1.7 .5 .2 .8 .2 .81.030 1.113 1.355 1.608 1.030 1.113 1.355 1.608 1.030 1.113 1.355 1.608 1.030 1.113 1.355 1.608 1.030 1.113 1.355 1.608 1.030 1.113 1.355 1.6080.073 .223 .479 .653 .051 .155 .333 .453 .020 .061 .131 .178 .009 .028 .059 .082 .004 .011 .024 .033 .001 .002 .007 .0100.003 .011 .037 .065 .002 .005 .018 .0310.004 .024 .0580.039 0.003 .007 .001 .008 .0200.175 .001 .002 .001 .003 .0050.67 .001 1.30 .001 .001 .001 2.42 5.15 Thevalues given inthetable aretheratios ofthealternating- current resistance attheindicated frequency andspacing tothe direct-current resistance. With large spacings thealternating- current resistance ofonewire isnotaffected bythepresence of theother wire. At5cmanappreciable effect isnoticeable, while when thewires arevery close together theincrease in resistance isseveral times theincrease atthelarger spacings. Ii8 Scientific Papers oftheBureau ofStandards [Voi. i6 The results given inTabte 6areshown graphically inthe curves ofFig. 3,inwhich therelative increase ofresistance with increasing frequency isshown. Atthehigher frequencies the resistance increases rapidly asthespacing decreases. 2.25 aitf FRDJUENCr Fig. 3.— Curves showing thechange ofresistance ofareturn circuit atdifferent frequencies Curtis] A.C. Resistance andInductance 119 TABLE 7.—^Alternating-Current Inductance ofaReturn Circuit [Length ofcircuit =1716.3 cm.No.2Cu.wire. Diam., 0.651cm] Spacing between wires in centi- metersFrequencyMeas- Com- ured puted induc- induc- tance tance micro- micro- henrys henrysPercent differ- enceInduc- tance; current distribu- tionofin- finite spacingValues ofeachterm inequation (79) First or log.termSecond termThird termFourth term 0.039. 0.175. 0.67. 1.30. 2.42. 5.15.500 1000 2000 3000 500 1000 2000 3000 500 1000 2000 3000 500 1000 2000 3000 500 1000 2000 3000 500 1000 2000 30006.64 6.30 5.73 5.18 7.99 7.71 7.20 6.86 11.24 11.10 10.79 10.52 13.90 13.78 13.58 13.27 17.03 16.94 16.70 16.47 21.42 21.33 21.12 20.896.88 6.70 6.34 5.71 5.29 8.11 7.98 7.70 7.21 6.86 11.34 11.27 11.11 10.81 10.55 14.02 13.97 13.85 13.61 13.39 17.13 17.10 17.00 16.79 16.59 21.50 21.47 21.38 21.19 20.99.9 .6 .3 2.06.88 6.85 6.77 6.58 6.39 8.11 8.08 8.00 7.81 7.62 11.34 11.31 11.22 11.03 10.84 14.02 13.99 13.91 13.72 13.53 17.13 17.10 17.00 16.83 16.64 21.50 21.47 21.39 21.20 21.015.16 5.16 5.16 5.16 5.16 6.39 6.39 6.39 6.39 6.39 9.62 9.62 9.62 9.62 9.62 12.30 12.30 12.30 12.30 12.30 15.41 15.41 15.41 15.41 15.41 19.78 19.78 19.78 19.78 19.781.72 1.69 1.61 1.42 1.23 1.72 1.69 1.61 1.42 1.23 1.72 1.69 1.61 1.42 1.23 1.72 1.69 1.61 1.42 1.23 1.72 1.69 1.61 1.42 1.23 1.72 1.69 1.61 1.42 1.23-0.15 -.41 -.84 -1.05-0.02 -.03 -.05 .01 .01 .02 The firstterm of(79) istheinductance oftwotubes. Hence thistermmaybeconsidered astheinductance caused bythe magnetic field external totheconductor. Thesecond term, identical with thelastterm of(23), gives theinductance caused bythefield inside asingle conductor. Hence thesum ofterms oneandtwogives theinductance ofthecircuit, assuming that thecurrent distribution isthesame asforinfinite spacing. The other terms show theeffect ontheinductance ofthechange in current distribution caused bythefield oftheadjacent wire. I20 Scientific Papers oftheBureau ofStandards [Vol. i6 InFig. 4therelative decrease ofinductance with increasing frequency isshown. Theinductance, Lq,atzerofrequency is always greater than theinductance, L,atanyother frequency, sothatfunction— j—-isinallcases negative. Asthespacing isdecreased, therelative inductance decreases rapidly. FREQUENCY 2000- 3W»- -.25 Fig. 4.—Curves showing theeffect offrequency ontheinductance ofareturn circuit Ihave received valuable suggestions from anumber ofmy colleagues attheBureau ofStandards. Also, Dr.F.W.Grover, ofColby College, hasreadthemanuscript with careandcorrected several errors inthenumerical coefficients. Dr. T.J.I'a. Bromwich, ofCambridge University, England, hassuggested methods forsimplifying theintegration incertain cases. Curtis] A.C.Resistance andInductance 121 VI.APPENDIX.— EVALUATION OFINTEGRALS ANDDEVEL- OPMENT OFSERIES Inthisappendix aregiven theevaluation ofsome oftheintegrals andtheexpan- sion ofsome oftheseries which arenecessary forthedevelopment oftheformulas ofthispaper. Ineach casethenomenclature isthatused inthebody ofthepaper. Only those formulas which arenotreadily found intext-books ofmathematics are included. 1.TOEXPAND LOG qINAFOURIER SERIES Thequantity qisoneside ofatriangle ofwhich theother two sides arepand s,having anincluded angle of6.Hence ^2^^2^p2_2j-pCOS9 Expressing cosBinterms ofexponentials q^=s^+72-7U +e )J p i9 I € S Taking thelogarithm ofboth sides ofthisequation logg^=log.^+log[i-^J +logi-^] Expanding thelasttwoterms andtaking theirsum 2logg==log.^-[^(^6^^+6-^^ +^,(e-^ +6--^^^ •. =2logSccos I 2S'cos20+~cos ^.6+ logq=logs—cos^—7—3cos2O =log^— i-r-j cosr93^'cos3^] 2.TODEVELOP ^^^ INAFOURIER SERIES OF Since and itfollows thatcosa+i sino;=qcosa=s—pcos6 qsina=p sin6 s—pcos6+ipsin6s—pe and q^=s^+p^-sp[€ +e ) But cosna+isinna=(cosa-{-i sina)"" cosnoi+tsmna s—pe s^+p^-sp n/id -idxU+e )i9\n 'K'-^) 122 Scientific Papers oftheBureau ofStandards Woi.16 expanding bythebinomial theorem andtaking therealpart 1'i^oL_ir cosnaiV nocosB.n{n+i)p^ 1^cos2d S 2 S^ nin -^1)(n-^2)p^« H^-^T^^cos3^+...- J 3.EVALUATION OFTHEINTEGRAL h Theintegral I^isgivenbythefollowing equation wheremandn areintegers. /i= Ipic?pir"pi"^log[p2+pi2-2ppi cos(^-6'i)]lcosw6'i(i^i Asshown above, thelogarithm termmaybeexpanded as follows: log [p2+Pi'-2ppicos{e-e^] ^/\r=2logp-2S j(-) COSr{B-By) ifp>p^ =2logpi-2S -i/-Y cosr (^-^1) ifp^>p When this issubstituted inI^,there resulttwotypes ofintegrals tobeevaluated. The firstgives thefollowing values: X27r 2logpcosn^idB^=owhenn>o =4TTlogpwhenn= Thesecond type ofintegral, viz: j"-2Si/"^'Y cosr{B- B,)cos^^, cf(9i maybeintegrated termbyterm asfollows: jcosr(B—B^)cosnB^dB^ =1/211cos[r (6'-^i)+n^i]+cos [r{B-B^)-nB,]\ dB^ =when r^^n ==ITcosnBwhen r=n Cur Its] A.C.Resistance andInductance 123 Dividing theintegration with respect top^,intheoriginal integral, intotwoparts, viz,from topandfrom ptoa: 7i=\'Px"^^' dp,j^L logp-2X^{^j cosr(d-dM cosnd,dS, +1Pi'^'^'^dpA \2logp,-2^U^ COSY{e-e^lcosne^dd^ 27rcosn0r pPi^+°+% C^ a. 71 1 =M pn^Pi+ Ip'^Px^^-'^^'dpA whenn>o =47r Ipi°^+^ logpc?pi+ Ipi°^+^ logpiJpiwhenn=o Allofthese arereadily evaluated. Below aregiven thevalues of/junder alltheconditions which may arise: Whenn>oandn—ni^2. J_27rcosnOr^^J1 i\p^a^-^+' 1 ^n |_^\m—n+2m+n+2/m—n+2j r 2p"^+2p^a°*-^+2-1 =27rCOSn^ 7 ;rz-0—7 ,x— l_{m-{-2y—n^{m—n-{-2)nj Whenn>oandn—m=2. whenn=o2irp^cosndV^, I1A= |_log p-loga--J ^^m+2 (m+2)2 Theintegral/aisgivenbythefollowing equation where g>P2 -^2=IP2dp2 I^og^'+p%-2gp2 cos{a-62)L"cosnd^dd^ Applying thesamemethods asunder I, h= IP2°''^'<^P2-—(") cosnee,whenn>o = Ip2°^+'c^p2(4 TTlog^),whenn=o 124Scientific Papers oftheBureau ofStandards [Voi. zdl cosna Substituting thevalues of—-—andoflogqasgiven above: I=-- +"("+'j,("+^)0Jcos3g+--..}] ^rr;.rz+>)cos.+^^(^(£Ycos..+. .1n{m+n +2)s^\_ \s/2\ \s/ J whenn>o 72=47r 1P2°^+^^P2Jlog^-^cos^-£3COS2 ^-^30033(9-•• -J = ;log^—-COS6 ;COS2d 1COS3^- • • ••m+2\__^ s 2s^ 3^^ J whenn=o. Washington, March 20,1919. iHALE TM)enHliEAt { miiMENGAHAa)MaiNthcyvane LTRVBR aeWe|ortee HOLA HR AIHalbaat BeaniesteWyneaRaeUthOrciae ATR eg|,ibe i iiWiHyehmerite PAR gisAMOtlyTt tid aps. ReneNET pa ; Denil)laMymanta)uh tieenimhtiihire AERA sa) Ve "