Snow_Circ544

UNITED STATES DEPARTMENT OFCOMMERCE +Sinclar Weeks, Serary NATIONAL BUREAU OFSTANDARDS +AY: Aan, Door Formulas forComputing Capacitance and Inductance Chester Snow ®g National Bureau ofStandards Circular 544 Issued September 10,1954 aS For byte SpendfDoman,U8GovePinOf,Wain, D.C rie10ce Contents Page Page Introduction......---ceececeeceeeeeeeeeeeee 1 2, Inductance andElectromagnetic Force—Con. 1.Capacitance.....2.2ececeseeseeeceeeeseees 2 2.5. Self-Inductance ofaHelical Wire... 32 1.1, Parallel Plates With Guard Planes... 3 2.6, Bifilar Matual Inductor.............. 32 a.Coplanar Guard and Electrode... 3 2.7. Coaxial Current Sheets... 34 b,Electrode atBottom ofHole in 2.8. Toroidal Current Sheets.......-...... 35 Guardeee 2.9.EndlessReturn-Circvits............-- 36 1.2, Spheres orCylinders...--e--ee---- 5 4,Concentric Cable..eeeseneeeeeenees 36 a.Concentric Case..eeeeneeeneeeeee 5 b.Two Parallel Wires (nonmagnetic)... 37 b.Plane With Sphere orCylinder..... 6 c.TwoParallel Wires ofMagnetic c.Eccentric SpheresorCylinders Material. eeeeeeeeenees 38 (Internal Case, 0<b<a,-0,)..... 7 4,TwoCoaxial Tubes......eeenennnee 39 4.Becentric Spheres orCylinders ¢.Two Equal Bars ofRectanguler (ExternalCase,62,423)... 8 Section. aeeeeeeeeeeeeeeeenees 40 1.3.Spheroids.....e-e eee 83.Frequency Effects... ---eevenceeenenee 42a.AThin Circular Disk ofRadius a. 8 3.1, Skin Effect inConcentric Cable...... 42 b.OblateSpheroid..... tees 9 3.2,Proximity EffectinParallel c.Prolate Spheroid......eeeeee 9 Wires... eeeecenenceeeeeeeeeeenee 1.4, Toroidal Surface....eeceeeeeeeeee 9 3.3, Single Wire Parallel tothe Earth.... 45 1.5. Conductor Bounded byTwoIntersecting 4.Legendre Functions That Occur inthe 2,Inductance andElectronngnetic Force...... 13 5.Derivation ofSone Formulas... 51 . 2.1, General Formation... 13 5.1, Eccentric Spheres andCylinders a,Axially Symetric Configurations... 14 (Internal) Equations (1.11) and b,Cylindrical Configurations........ 17 (LID) sesecsesceceneceseeeeeeeeenes 51 2.2, Circular Filaments andCircular Turns 5.2, Eccentric Spheres andCylindersOfWire.eeeeeeeeeeeececeeeeeee OL (external) Equations (1.14)and8,CoaxialCircular Filaments... 21 (21s)ae ».Circles Whose Axes Intersect... 22 5.3, Derivation ofEquations (1.17) and ©.TwoConcentric Circles (Not (1.16) forOblate Spheroid and Coaxitl)..ceeneneeeeeececceese 23 Gircular Disk... 58 a.TwoParallel Circles... %4 5.4. Derivation ofEquation (1.18) for e,Self-Inductance ofaCircular Turn Prolate Spheroid......-..s--ennene 56 OfWire. neeeceeeeenene 26 5.5, Derivation ofEquation (1.19) for@ £,Self-Inductance ofaCircular Turn Toroid..---e-eeeeneeeeeneeseeeees= ST ofWire Near aMagnetic Medium... 27 5.6. Self-Inductance ofaSingle Turn of &.Self-Inductance ofaWire... 27 Wire, Equation (2.15)... 59 1h,Matuel Inductance ofTwoParallel 5.1. Derivation ofEquation (2.16) for Wires Having theSone End-Planes 28 Self-inductance ofaSingle Turn of a.Matual Inductance ofTwoParallel Wire Near aMagnetic Medinum......... 63 Wires notCo-terminous........... 28 5.8,Derivation ofEquations (2.40) and J.Mutual Inductance ofTwoEqual (2.41) fortheSelf-inductance of Rectangles Lying inParallel Toroidal Current Sheets (Tepe Planes.....--enenceeeceneeeneeee 29 Winding)... ceceeceeeeeeeeees |OF k,Self-Inductance ofaRectangle... 29 5.9, Derivation ofEquation (2.45) for 2.3, Concentric Solenoids (current Self-inductance perUnit Length of sheets)...eeeteens 30 ‘TwoParallelWiresofMagneticMa- 2.4, Self-Inductance ofaCylindrical terial...eeeeeeeeeeeeeeeeeeee 4 CurrentSheet-o-.e-eeeeeeeeeeeeee 31 6,References........2eeeeeeeeeeeeeeeeee 6B ay | : : Formulas forComputing Capacitance| andInductance Chester Snow Explicit formas aregiven forthecomputation of(1)thecapacitancebetween conductors having agreat variety ofgeometrical configurations, (2)theinductance, bothself- andmitual, ofcircuits ofvarious shapes, and(3) theelectrodynmnic forces acting between coils whencarrying current. Form lasforskineffect andproximity effect inconcentric cables andparallelwires areincluded. ‘Theformlas forthe‘simpler configurations aregiven in terms oftheelementary functions, whereas morecomplex shapes involve theuseofLegendre polynominals, Legendre functions, andelliptic functions. One section isdevoted toadiscussion oftherelation between theLegendre andtheelliptic functions. Introduction Thiscollection offormulas contains somethatarecommonly usedinelectrical work andsome thathave been specially developed forprecision work atthisBureau. This isno attempt atcompleteness, forthere isnowavailable (since 1948) revised third edition oftheearlier compilation offormilas forinductance byRosaandGrover [1].! Thismay beconsulted forreferences tooriginal memoirs andalsofordiscussion ofthemostsuita- bleformula for»given configuration orrelative dimensions. Reference mayalsobemade toDr.Grover’s [2]additions tothese formulas in1918andtohisbook"Inductance caleu- lations working formulas and tables.” Formas forcapacitance maybefoundinthesecond edition, 1924,ofaworkbyJ.H. .Dellinger, L.E.Whittemore, andR.S.Quld [3]. Thiscontains formulas forinductance andafewforcapacitance. Itispossible thattheaggregate ofresearches oncapacitance uptothistimemight amount toacollection ofcapacitance formulas ascomprehensive as that ofRosa andGrover forinductance. Theformulas given herecontain, inaddition toelementary functions, theLegendre polynomials P,,theLegendre functions 0,yandP,yandelliptic functions. Itisshown insection 4howthelatter twomaybefoundbyuseoftables ofthetwocomplete ellipticintegrals Xand 8. These,together withtheincomplete integrals F(d,k) andB(¢,k), andanu,cnU, dnu,ete,maybereadily foundfromthe1947Smithsonian elliptic functions tablesby G.W.andR.M.Spenceley [4]. Qnepoint ofsuperiority ofthiswork over that ofBR.L. Hippisley [5]isthatitproceeds byincrements of1°inthemodular angle instead of5°. Another purelymathematical tableofelliptic functions andthetafunctions thathas beenfoundveryusefulistable1,(1922)byH,Nagaoka andS,Sakurai [6].Thesameau- thorsin(1927) [7,table 2]produced avolume moredirectly applicable tothecalculation 1Figures inbrackets indicate theLiterature reference attheendofthepaper. 1 ofthe force between coils and their self and mutual inductance. For the latter, Nagaoka [8]has also published three formulas that make use ofthe remarkable convergence rate of the series defining the theta functions inpowers ofthe Jacobian paraneter q. Short tables oftheta functions are given byJahnke and Ende [9]. Also short tables ofF(d,h) and E(¢,k) were given byB.0.Peirce [10]. The recent work ofW.Magnus and F.Oberhettinger [11] isvery useful. These volumes, especially the work ofthe Spenceleys put the computation offormas with elliptic functions inquite adifferent light. Such formulas are not more difficult than those with sines, cosines, and logarithns. Insection 5are placed afew notes onmethods ofderiving sone ofthe formulas given here that are not generally available, orperhaps are unpublished. Where space permits, ithas been attempted tosummarize the entire électric field, onwhich capacitance is based, orthe entire magnetic field underlying the inductance constants. Such ascheme seems desirable onalarger scale than ispossible here. Fach formla for capacitance re- quires the evaluation ofthe electric potential orfield atevery point ofspace. For each inductance Zor4,one must find the vector potential ormagnetic field everywhere. The constants C,L,orMrepresent asmall byproduct, since they are derived from the fields bydirect processes. Asummary ofthe more important electric and magnetic fields that have been evaluated todate would probably fit present requirements better than fur- ther tabulation ofcapacitance and inductance. 1.Capacitance Theformulas forcapacitance given inthispaper areexpressed inthecentimeter- gram-second electrostatic system ofunits (unrationalized). Iflengths incentimeters are substituted forthecorresponding symbols inaformula, theresulting value ofCwill be thecapacitance incgselectrostatic units. This value should bemultiplied by10/9 (more precisely 10/c =1,11277) toobtain thecapacitance inmicromicrofarads. The formulas as- sume adielectric constant ofunity (in the egs-esu system). Ifthe space between elec- trodes isfilled with adielectric ofpermittivity ¢,relative toempty space, thevalue ofcapacitance ascomputed from the formula should bemultiplied by¢,. Alternatively, when expressed inthe rationalized meter-kilogram-second-ampere (Giorgi) system ofunits each formula for capacitance would have anadditional factor of 47(byreason ofthe rationalization) andalso afactor of10'/4nc? (byreason ofthe conventionally chosen permittivity offree space). The net result isthat with the dimen- sions expressed inmeters, andafter multiplying bythecombined factor 1.11277 x10", the resulting value ofCisinfarads. : The formulas for inductance and electromagnetic force given inthis paper are ex- pressed inthecentimeter-gram-second electromagnetic system ofunits (unrationalized). Inusing the formlas, lengths should beexpressed incentimeters, currents inabamperes (i.e., units of10amperes), and thepermeability ofspace should betaken asunity. If this isdone, theinductances ascomputed areinunits of10~® henry, forces areindynes, and torques indyne-centimeters. Alternatively intherationalized meter-kilogram-second-ampere system theformlas should bemultiplied by1/47 (byreason ofrationalization) andby47-107" (byreason of 2 |theconventionallychosenpermeabilityoffreespace).Thenifdimensionsareexpressed inmeters, andcurrents inamperes, the inductances ascomputed will beinhenries, the : forces innewtons, and the torques innewton-meters. : ‘Thefirst sixfigures illustrate twocases. Intheaxially symmetric cases thefig- ures represent plane sections through theaxis ofsymmetry. Inthe cylindrical case they : are plane sections perpendicular tothe endless generators. Inthis case the formlas ; give thecapacitance C/lperunit length perpendicular totheplane ofthefigure. 1.1. Parallel Plates With Guard Planes ‘The separation cbetween theparallel plates should besmall compared totheradius @,ofthedisk. Also theradius Aoftheplates should belarge compared toa,,sothe field ispractically uniform atsome place between theedge ofthedisk and theouter edge oftheplates (4>5a,). 8.Coplanar Guard andElectrode [13] LY)CLG A Aina VeVy| [¥49 electrode’) xis Froure 1 aUf <1 A 1@=5(a,+a,), and(a,~a,)/¢ issmall. Axialsym: ont?7d(2701)oth, ger (ast) cob. (a) indriG@ lfay-a,\2 wa Cyclindrical:°21 (2-0)1)coth™, . 55877 |“Fre Ne 0.2) Thiscapacitance isbetween theplane atpotential 7,andtheelectrode, including its plane face and its sides. j : 3 b.Electrode atBottom ofHole inGuard [13] (Capacitance Between thePlane at7,andtheFace ofElectrode atBottom ofHole) ‘Asinthepreceding case,a/Amustbesmall(eg ).Alsotheholeisnotvery shallow, and the clearance between the electrode and its guard isignored. da el 1< ©, and O<y= <2.pofeo, and0670 i 4 : Ay 1a vey, +00 Yjeg.—4-doxis Froune2 c V=07, Axialsym: ont Jina (1.3)TZyadsinha,(Pty)or where a,=2.4048, a,=5.5201, @,=8.6537, a,-11.7915, andJ,(a,)=0. Thefirst three terms aresufficient, withtheconditions givenabove, foranaccuracy of1in200.GW,is Bessel’s function). . Say]41g2"=1sin-! Cylinder: o/1=89 (=1yntig?e} sin(2n-1)¢ a.4)7 (2r-DC1=(q?)4) or where ~ 7qeloe) [usecost6e-4(«F)] ea420-8 sin6cosoe-+(2 ) a =tan~! = af adians).e=tan-t (1/7) stan(2)(@inradians) Incase thehole isvery shallow (d/c small), abetter formula than (1.4) forthe cylin- drical case is eo(ay[(2)"*] aise4oo)\(¢(ar) ; c/a -+fig1YX" -___-}, as) Ie"ama (ca, nlltai?*] on 4 where randq,maybecomputed by logren™? - © tog£f1-£271 coth(ma/e))Bree ya asink(ma7ey 1+€coth— 2 er(1-Jeoth(na/e)) logq,=log—— 2? $61h08 2(1=r8) where e=d/c. Equation (1.4), Like (1.4), isexact with slotofanydepth. Both ignore clearance hetwoen electrode anditsgurd. Totakeaccount ofthis(tofirstorder) let2a,denotethewidth offaceofelectrode; 2a,,thewidth ofslot; andd,itsdepth. Thenif B=(a,40,)/2, z 7 4csinh1a/c oftageh(=)tant(six)Soviet |a4 re*Sr G a-a, )72+(4,2) neglecting terms oforder {d2+(a,-a,) #]logld?+(a,-a,) 2). 1.2. Spheres orCylinders #.Concentric Case ° & » y| fe ‘ Frouae 3 :=2192 Spheres:ongate (1.5) Thecapacity ofonesphere alone (a,~0) isC=a,. L Cylinders: c/tz 1_ (6)a 2log.22 a 5 Equations (1.5) and (1.6) are limiting cases of(1.11) and (1.12). The potential between the spheres is ey 3xa2/r nn that between the cylinders is _)loga2/r TOTToge/a,” b.Plane With Sphere orCylinder (This isaLimiting case ofequations 1.14 and 1.15.) g|>| R d xis Froune 4 {aae] ntVintaa? y=?tog(MEER). (1.7) Hyy-inthly Sphere: c-2Vara? SS (1.8)ho1o-lw ly Cylinder: on=-——1_ 5° a9)79logLEVRI=oFa When h-m, y+log (4h?/a?) and (1.8) gives C~a, but(1.9) gives C/1=0, asitshould, since thelogarithmic potential becomes infinite atspatial infinity foranyfinite charge except. zero. 6 | ¢.EccentricSpheresorCylinders(InternalCase0<b<a,-a,) 2toy, |& .loxis Frcune BxB,>By B-B,>0 b-00, 0,0,=b= thedistance between centers (always positive) 2be=V[(a,4a,) =b*][(a;-G,) =0](positive) 2 g?p?A,=logseepeete (positive) (1.10) a}-a34b?42be . By=logOe (positive) =~(2nt1)8, Spheres:c=ze Ye a.)a(n A,-By)=o Ine 2yqt-p? 1a}ta}-b? 42be linders:C/t= 1, og——~—_____ (1.12) Gjlinders vtTB=By)feoea,a, 7 d.Eccentric Spheres orCylinders (External Case, b>@,+a,) UZ 3 Ly SyRY Fieuae 6 B=-B,<0 B=B,>0: >-oB b=010, 2de=V[b* (a,4a,)*JLd*-(a,-a,) "J(positive) tol-ofstbe) ces =padTksBeOS q ve Byroel!Ta,b (positive) te etaataan(1.13) _0?-af+a} 42be iti By=log(pesisstee (positive) b?-ai~a}+2de . 7=2(8,48,)=2 log] (positive) 2) nlkly 2 heres: c=2e ~~ 4 (1.14) Spheres: =tne 3Y . hene! 1-6 Sesinh Cylinders or L 1ra 3: O/1s” (1.15)perallel wire: /FIG) Placing b=a,+h anda,~, eq(1.14) and(1.15) gointo eq(1.8and(1.9), respectively. 1.3. Spheroids a.AThin Circular Disk ofRadius a o=*2, (16) 8 i |This isalimiting caseb=0, ofthefollowing formula, | b.OblateSpheroid ) Majoraxis2a,minoraxis2b: Vata? c=Ve)" (1.17)5)sin(2° a | c.ProlateSpheroid Major axis 2a,minor axis 2b: Vatapi c=—rat-d? (1.18)le(@ia=y)eV 1.4. Toroidal Surface ding CF)exisof-—-b--rotation Froune7 a=radius ofgenerating circle 4=00>a 2 Acosh6,=74-15 nt=2e80O<k<1 AVROT 7Opnxleoshhy) eal sab,cyleoahA.)(£07Y2»En}ifm0). (1.19)oe PandQarethetwoLegendre functions withthesameargunent 4/a.Amethod offindingthese functions fromtables ofelliptic functions isgiven insection 4. 9 1.5.Conductor Bounded byTwoIntersecting Spheres [14] (Alone inspace) LS ~ eo 7 Figure 8.—Axial section of intersecting spheres. c-radius ofarc onthe right, semiaperture=@ a=ie ofarcontheleft w=angle atwhich the arcs intersect. | Allfiguresmaybeobtainedwiththerestrictions (O<6<m and O<w<27. |‘Thecapacitanceforgeneralwand@isC,(9),where | asing( 7 nm6 1 nm 7ogra’ {HeSate v(md)-» (Foz) og]. (=F re sin? |v(me)-¥(ty)Boe (1.20) sin where ydenotes thepsi-function, F/I, whose values maybetaken fromthetables ofH.T- | Davis, "Tables ofthehigher mathematical functions" (Principia Press, Bloomington, Ind., 1933). Theseries (1.20) converges likeZ1/n?, Foramuchmorerapid series converging like | 5n-14, seereference [21],wherethecasesareconsidered thathavefinite termsforca- \ pacitance(u/m rational). \ | ‘ | |10 |a i | ‘Thesimplest oftheseisthelimiting case»=2n, wheretheconductor isathinshell 1 with any aperture 26. raaX a.oNts Figure 9 radius o ween eae C(6)=a-5(6-sin 8). (1.21) |— For@-n/2 thisgives thecapacitance ofahemispherical bowl 11. C,,(1/2)=a (ie)=.8183a. Orthogonal Spheres: (External, v=7/2) | oe ~ .—— wy, ; aa, OppOCstaneosinBata, Fae (1,22) since @,=0 tan 8. ll Orthogonal Spheres: (Internal, u=37/2) we Froure 1 aK sin6| 1 o¢e)AaV5-$+ zLat—e7 ea (1.23) aunPain’in’nine2084(cosS+sin2) 3 3 3, 3 3 3, where a,=a tan@inthis case also. Hemisphere: (w=37/2 and 0=7/2) ‘AHemisphere Ficune 12.)(watKonaBayz)Ue Placing 6=n/2 inpreceding case gives Cyy/,(n/2)=20 (1-75)=84530. (1.24) More generally, for w=7/m, where m>1, a 0(6)=ata sin6 —e (1.25) ve 1 \ asin(2:2) sin ro and for w=27/m, where m>2. _ 206 at 1-(42 wt 0(8)=a-2(6-sin @)+asin6Gs)_(-F) (1.26) aea(20) (2) 1 at cn Or) C,(6)=a (complete sphere). 12 . , There are finite suns forcapacitance when v=(FET)a»,here1eneon, 2R and vagbtin,whereLcnc2m-l. 2.Inductance and Electromagnetic Force | 2.1,General Formulation Iftheunit oflength isthecentimeter andthepermeability oftheconductors and thesurrounding media areunity, theformulas below giveinductances incgselectromag: netic units, thatisin10-9henry. Iftheelectric currents ‘I,and'I, areincgselec- | tromagnetic units, oneofwhich is10amperes, theelectromagnetic force isindynes and torque indyne-centimeters. 1 Thevector Bofmagnetic induction isthecurl ofavector potential 4.Ifawire of appreciable cross section, andintheform ofaclosed circuit, carries aunit current whose volume-density isthevector {,,theintegral overthevolume ofthewire Sffcane, isascalar quantity which iscalled either theself-inductance ofthis current distribu- tion, oritsmutual inductance with thefield, according asAisproduced bythis distri- bution alone, orentirely bycurrents other than itself. Ifthemagneticpermeability is1everywhere, thevectorpotential 4atanypoint Pduetoaunitcurrent inawireNo.2,whosevolume density ofcurrent atP,isthe vector t,,is taav ©{SfgeRr integrated over thevolume ofwire No.2,where Risthescalar distance from thefixed Point P,tothepoint ofintegration inthevolune-element dv, Themutual inductance ¥between thetwocurrent distributions istherepeated volume integral dv o[ifo Ifooe R 13 When the wires shrink tomathematical, closed curves this becomes Neumann’s double line integral ds,1fas,feos(as,.40,) 9 taken completely around both curves. The self-inductance of aunit current distribution in awire is (ito. Ieavf|Coav. Sellfe Inthe case oflong straight wires, the formulas below apply forthe uniform current distribution. Inthe case ofwires inthe form ofcircular turns, orinform ofhelices, the few very accurate formulas given below apply tothe"natural" current distribution (current density inversely proportional tothedistance from theaxis ofsymmetry). ‘The distinction between uniform and natural distribution isonly ofinterest for pre- cision measurements. Inthe formulas forLand ¥tobegiven forparts ofaclosed circuit (such asLfor along straight wire alone, or¥fortwoparallel ones), these expressions must beunder- stood torepresent only such contributions tothemultiple integrals forLand¥asmaybe written without specifying thenature ofthereturn circuit whose contribution is,of course, tobeevaluated bythe same type ofintegral. ‘Themajority offormulas forLand¥that aregiven below fall into oneorother of twoclasses, ineach ofwhich theabove volume integrals arereduced tosurface integrals over aplane section ofthe conductor. a.Axially Symmetric Configurations ‘The first class isthat ofaxially symmetric conductors forwhich the surface inte- grals aretaken over across section inaplane through theaxis ofsynmetry, saythe x-axis, where (x,0,¢) arecylindrical coordinates. Theonly component ofthecurrent den- sityvector is{4=t(x,p), independent oflongitude .Theonlycomponent ofthevector potential AisA,=A(x,p). Thecylindrical components ofthemagnetic field arederived from AbyBoyuf=curl A,so a,=1p,(eA(x, 0)He=2D Led (Xs nll=-D,A(x,p) andHy=0. With »=1 everywhere (A) LorHis2nffpit(x',p') A(x’,p')ax'dp', u“ where Jftcxoraxdoer (unitcurrent). | ‘Thesurface integrals aretaken over anaxial section inthe(x,p) half-plane. Thesymbol| has beenusedtodenote thedistance (inspace) between twopoints P(x,p,¢)and \ P'(x',0',4'). Wemaydesignate by this distance when thepoints areinthesame axial | Plane (¢=¢") ; RIE(xox!)24(p=p')? | BR?! RECA)pte"t-29p'condo")=200"[Itg8X-cos(é-4")]« i | There istheknown Fourier series ‘ 12' RP B: SS nn(15) osn(¢-$'), ;wD onlas |we |where€4="/,and¢,=1ofn>0,andQ,.yisaLegendre functionofthesecondkindwith parameter n-¥/,. Itsreduction toelliptic integrals isgiven insection 4. Equation (B) isequivalent to "cosn(d-$') 2 Rt (Bysonrenea=ex(its : |[Ropel55 | Sincetheonlycylindrical component ofcurrentdensityis14=1(x,) independent of 4,theonlycomponent ofvector potential willbeA,=A(x,0) independent of4.Hence, it would besufficient toevaluate 4for¢=0. However, thevolume integral defining 4isthevector equation fjfe. ff ftglx'e'de! A=———_—_: pidp'ax' |——————_.R 1,OR Thisintegral isthesumofmanyvectors thatarenotparallel, sothatwemayuserec tangular coordinates, y=p cos¢andZ=psin¢d, andwrite Lyantg(x'p')sind’ and1,=t4(x',p')cosd’, 15 20 (Pott. paotaxt[aneiad Ay(2)ffetle",p')dp'axii ,+ctyaergy["cosdo!de’ Ac.o=[fo tx",p')dp'ax[wa Hence 1” cosld-' dad"4gzA(x,p)=-4 (xp)sind4d g(ap)eoed=ffo"Hx",2"Men"defeetay. R Consequently, byeq(B’) with n=l, © Ag=A(xZhVa"U(x",0')0(ree) ay' © eA ed=yg] [WoT 010s Soa? e's so that 4satisfies 1 1 ~47t(x,p) inS ,ayptety 1) a= ©)(o2+0320-33) (Geiaes The integral in(C) istaken over any plane axial section Softhe conductor, which may be ofany shape. Its self-inductance Listherefore Oo) L=20 t(x,p)*p(x, p)dxdpIf nx!)?pee") anf[tex,praxdoff0100's9'904(1422) crap’, I, I, 2pp where Jfree.praxapai. Also from (C), the matual inductance ¥between two coaxial wires with any shapes or size ofaxial sections S,and S,is x (xx)? +p,-P2)” CB)amar]|oftOy,eydxrdp, ||oftOey,02)0y{14+ Jax dg, Pye 16 ¥ where | fftsGareddesdoy= f[teOrg192)axqde, al. Letting both sections shrink topoints gives (Cerra?H-)") (F) H=4nVa, a,0ya asthemutual inductance oftwocoaxial circular current filaments ofradii a,and ay, in theplanes x,andx,(eq(2.1), page 1). See section 4forthe evaluation ofthe fune- tions 0,-y interms ofelliptic integrals. Ifthe section §shrinks toapoint, eq(C) gives the vector potential A(x,p) atany point P(x,p) inspace, that isproduced byunit circular current ofradius a,inthe | plane x=0, and coaxial with the x-axis 24(p-a)? -F 40,0)°2//40y(1-o ee”®)2222 -na], 2apol where pte 400. x*+(pta)?? keisthemodulus ofthecomplete elliptic integrals Xand F. Thecylindrical components ofthemagnetic fieldaregivenbyH,(x,p)=1/pD,(pA) andHx, p)=-D,A, 80 Bepele Ee] Veiorayih Hamp) 4ox Len a 4H,(x,p)=yp En (EF (xe)orca 2(5)]- which apply for any point (x,p). b,Cylindrical Configurations Theother class offormulas maybedescribed (with rectangular coordinates x,J,z) as applying toaunitcurrent withuniform current density {,=1,=0, £,=1(x,y) (independ- entofz). Thetotal current +1flows inthefirst conductor with cross section S,its density being 1,=1/S. Thereturn current density inthesecond ist,=-1/S,. Theself- inductance "per unit length oftheline" isdenoted by1/1, which means theself-induct- 7 ance oftwo cylinders having the same two end planes, these planes being separated byone unit oflength. 1 1w/t=zffas-bJfees, 8, ‘Ss, , where Aisthe potential ofboth distributions. When 1=1 everywhere, the value of4at anypoint xyintheplane is(where 4denotes 4,, the only component ofA) 4=22{[togras,+2 Ras Comas ogas+e logRdS,, where Risthedistance from apoint ofintegration P,indS,, (orfrom P,indS,) tothe general point P(x,y) inthe same xyplane. This gives w/=Zffasffroenas;-2 [fasfflogRaS344JfesJfroezas: 3 1 aH 2 vas, 1 2 The first two integrals are frequently designated by1,and L,, respectively, the third by =2Myq, buttheseparate integrals only have ameaning with reference tothis equation ofa "closed", orreturn, circuit. With this understanding, theself-inductance 1/1 per unit Jength ofthe line iswritten (6) L/1=-2H yapthyy+h9/1=2C2logDy_-log Dy,-1og P22] where logDis==yffiesfflogRdS.08 Dyy=oe og'S8,JJs, 1IIs, , CC) 1 logDy,=— [{48, logR4S,,8,7 Ns5, zi and similarly for Dy». ‘This D,,defined bytherepeated surface integral over thetwocoplanar areas S,and S,iscalled theg.m.d., orgeometric-mean-distance, ofthearea S,from S,,andD,, the geometric-mean-distance ofthearea S,from itself. Thedefinition isconsistent with the extension tothe g.m.d. ofone curve from another orofthe g.m.d. ofthe line from itself. Inthe case ofareturn circuit oftwo parallel wires whose circular sections have radii a,anday,itiseasily found that aw logD,,=log a,~Y4, logDy=log a,-%4, 18 and (ys) logD,,=log b,wherebisthedistance between centers. Thegeneral formula (G)fortheinductance L/1 perunit length oftheline, 4/1=2[2logDy,—LogDy-logDy;]giveseq(2.44). Foratubular conductor whosecrosssection isanannular areawithinnerradiusa, andouter 4,, itisfound without difficulty that a,? 2a,? mn)a]ay (Kk) JogD,,=10g4y-——-_——— |——. log] -1]-,a 20,2] Aytaay? NT Whena,~4,,thetubebecomesinfinitely thinandthisapproaches thefinitelimit logDy,=1og 4,sothat4,istheg.m.d. oftheperimeter ofthecircle fromitself. Whenthecurrent intubeNo.1returns inalargertube,No.2,coaxial withit, whoseannular section hasinnerradiusa,andouterAy,itisrelatively simpletofind thattheg.md. between thetwoannule isgiven for0<a,<4,<a,<A, by 1 2 2 y wlogDy,=———A log4,-a3 loga]~%Aj-a}7%] j Formula (G),£/1=2[2 logDy,~logD,y—1og Dz]inthiscaseleads toeq(2.46) by' (K) and (L). Inchecking fornumerical errors, itmaybenoticed that theformla (G)forthe self-inductance perunit length ofaline (areturn circuit) will bedimensionless. This } isnottrueoftheconstituents logD,,andlogDjs,etc., astheydonotinvolve logarithms ofthe ratio oftwo lengths. Ingetting g.m.d., usemaybemadeofanyformal analogies tothelogarithmic poten- | tials ofelectrostatic distributions, forthesanefirst integral occurs inbothproblens. i Thelogarithmic potential ¥ofanendless cylinder ofanycrosssection §withunit |density perunitlengthperpendicular totheplaneofSis } Vx,r)=2 fftogRax'ay’. Forexample, whenthesection Sisacircular area, oranannulus, Yisthesameatout- sidepoints asifthecharge wereallconcentrated atthecenter. Atpoints inside, a simple lawprevails. Thismaybeusedtocheck theg.m.d. given above forcircles and annuli. } Whenthetotal charge Qperunitlength infinite space iszero, thelogarithmic potential (like £/1forareturn circuit) isdimensionless astolength. Hence theca- Pacitance perunitlength ofendless cylinders, asineq(1.1) to(1.15), involves loga- rithms oftheratiooftwolengths, Butthepotential ofacircular cylinder withcharge i Qhasthevalue 20logratoutside points. This isnotdimensionless, 19 Afinal example {15}maybequoted, theg.m.d, ofarectangular area from itself. If the width of section is wand breadth b, 2 242)? Day? 5)=4logD,,=-2log(w24b?) 2log(SE)%tee("2")3b? w? 3w? ob BY gag? gant¥)425 “alo "te b)t3 rr 2 oerts)854togweed, ‘3523 3 where §=b/w, and a P(x)=-2x4 Logx¢(1-6x24x4)log(14x?)-8x(1-x? )tan“tx, Since theexpression forlogD,,issymnmetric inbandwv,this suggests theidentity inxthat iseasily verified. (0) Ponsxte (2)-1207Logx-4nx(1-x?), sothat, from apower series for F(x) inpowers ofxvalid, when x<1, weget bythis, the series forx>1.Theseequations, (M),(N),(0),areusedinderiving eq(2.47). Iftwo ormore long parallel, cylindrical conductors all carry current inthe sane direction with the same uniform current density, they are effectively one conductor of cross section S.Ifthesections arethecoplanar, nonoverlapping areas $,,8,,,, etc, thenS=8,+8,+5,..-+ Forexample, withthreesuchsections theg.m.d. ofthecompound area §from itself isgiven by ) (5,#8,¢5,)? 106D,=ffasffoenas’ =Sf.[frrasisffs.ffrosRas$+ff4s,ffloenas, +2JfesifPreeRAS+2SfeSf logRaS,+2JfessffreRas, “postneyeepetyen8syeptateypalsyenatePlog att )-«D92(D58(DMEyaHeyyMast), ‘The generalization ofthis isnot difficult, when each conductor carries auniform but different current density. Weighting factors are introduced. 20 2.2. Circular Filaments and Circular Turns ofWire a,Coaxial Circular Filaments [16] i Oy a ! Figure 13 : La Their mutual inductance #is weanyor,[2ar]anyon,(2-1), (2.1) where themodulus &ofthecomplete elliptic integrals isgiven by 40104 (2.2) naAO, - Mery(a,40,)7 Their force ofattraction isY=-I,I,3¥/2x or LiTaxk P2ektTat, qyonF2k|- (2.3) ' 2 b.Circles Whose Axes Intersect [17] Ae Ay Zei M+cose,(0£4Sx) °Zar byscosa,(OL,<7)percos@ (059 £7) Froune 14 ear (Lau2)(deut) $7TELEPar)PaCuePal) wearin) Oe)7ToeSA eran pe (2.4) =40?(1-n3) (1a) TN Palas) Par Pa) igror, rt n(nt1) = whatever thevalues ofa,/a,. P,(y:) istheLegendre polynomial andP;()=(d/du)P, (u)« Inthespecial case where thecenter C,ofcircle 2coincides with 0,r,becomes 2, anda,becomes n/2, and(2,4) reduces to 3 a, "PC x)wear?a,(1-ut) >(-1)"(2)TP banHyPene()- (24) on The torque facting oneither circle, tending toreduce theangle @between their axes is On aM, 3/, . = nfo2\ 2DUH) .ro-Besinopt-an®/Paysin (I-ut) (1)(=) THePineUy)Pian(H+2.5) oa These converge forallvalues ofr,,ifa,<a,. Mis positive when thecurrents circu- lateinthesamesensearound theiraxesOC,and0c,. 22 c.Two Concentric Circles (Not Coaxial) (Special case ofeq(2.4')) Ay A ae 3 rodit -0A . “eo adive 0+OA, HecosB agOR, op<0, rk 9/22) nse) P(nt%)=— —) meen tu) *a,>(2)mi(nai)? Pann+ @6) me pte 22nAKIEA)9+(yy Q.7) a, ai) ni(nely) Pant = Axes Parallel (6=0,u=1), 1=0 \ yt 92N2"PCntK)(nt¥) (2.8)| ay ay ni(ntl)!* i oa i} AxesPerpendicular (@=n/2,u=0) N=0 petVOE9724)2OCn4K)[ey]. 2.9)~ ay a,/ (nei)! nt . or 23 4.Two parallel circles [18] Case where 0<r<a,-a,+ posso i —a-he 4,ale “yee B=005.8 or cunt16 A 22"OA, Anat 7 ,ye? tor(z) SALLERe (2.10) aD anl me where F,=F(nt'4,nt%,2; a7,/a7,) (hypergeometric series). Ifthecircles are coplanar, (8=7/2,4=0) and pla)(1)xDOnt#)nt Case where r>a, +a, aaa A :rr fe N\, * an) 1. aro, ‘4ree 4\ pwrcos8l 1 a \ r=00, Froune 17a | 05K,0URE /a /0,7OLA, / J / Anisat 8y\241Pn4K) ->Yop(2 2.11 a» a, (-1)")TnatytPnPan(4) () oT where F,=P(-n,1-n,25 a/a?). 24 Forequal radii a,=a,=a andr>2a =. 2a\**1 PCnt)PUntk) a aay(29 I4) . W=2na>(-1)(2)Teah)pty Pen(4) (2,12) or For twoequal circles, coplanar andexternal, this becomes 1/2) 2001 T3(ntk) =? [2a $) 2.13) "“oy()GeDtatGa! c cn (The plus sign would apply when thecurrents circulate inopposite senses with respect to thenormal totheir plane.) Thetorque iszero. ‘Theforce ofrepulsium along their line ofcenters isF=+I,I,2M/2r, or S41 /2.0\ Put %a)P?(4%) PotntyTy>(2)Ginity? (1Mor 25 ¢,Self-Inductance of aCircular Turn ofWire 3 A P=distonce ofopointfromxoxisae jo xa=rodius ofgenerating circle Froune18 A+OC Garrent density ofthe unit current is zliey’‘ezarp(Z) Gar)» where _p(Bled 2): d(b-1) a? PoP(Pop2iga aSEto2dorder a? 847(b-1)(d-*4) a? (34) j=4rd — 0g—=+——————_ —,log— }. 2. teawa- [14(2041) |togSAT,SOME?Fysero(5lows (2.15) For uniform current distribution b=0. For “natural” current distribution b=-l. For b:=5/2, themagnetic field outside thewire isexactly thesame asiftheunit current were concentrated inacircular filament ofradius VY4?-a?, 26 f,Self-Induetance of Circular Tarn ofWire Near aMagnetic Mediua a 20 Z +l it ja |ijo_#*! BO)Prous19 (oir) | uel 2 =,—4m:=- |ears ts404(21) ! bel[ue ] =1,,+tana[20-2 eg] - eta” A % RE (2.16) 4Aa ‘modulus k)where x2=—442__ (modulus8)whereb=ay 4,;,maybecomputed bypreceding case, (2.15) onthebasis z=everywhere. Incorrecting theself- ormutual inductance ofcoils fortheeffect ofthin lead-in wires, thediameter ofthewire isimportant byaffecting itsself-inductance, butwires maybetreated aslinear conductors inestimating their mutual inductance. | &Self-Inductance ofaWire | Vitya? 1 [=2[1dog(HPeette)_Virxaisisa|: (2.17) whereIisitslengthandaitsradius. 27 h,Mutual Inductance ofTwo Parallel Wires Having the Sane End-Planes ae D Froune 20 +L ——_________-. ny Se Vir DiI 2[Llog(422F Vimeo | (2.18) where Iistheir length, andDthedistance between centers. Ifthecurrents are inoppo- site directions, the sign of#isreversed. i.Mutual Induetance ofTvo Parallel Wires Not Co-terminous Cy ~~D be----e----+| L a Ce Froune 21 RA -s B--ts--—--4 ©*GC,where C,ond C,orecenters ofthewires -butts) (uth) (beh). (tt) (2.19) sew(c+Ft)tw(c--2t 2)-w(oro )-w(e+ ), . where avx[eltog[eels!) _yee, 0that w(x) isaneven function ofx. This holds for collinear wires (D=0) ifthey donot overlap. ‘The sign of¥isre- versed ifthe currents have opposite directions. : 28 J+Mutual Inductance ofTveEqual Rectangles LyinginParallel Plasce (One isthe perpendicular projection ofthe other.) The distance between theirplanesisd,thelengthandbreathofeachisaand |b,respectively. | Neunann’s formula is |®ato,[(a+Vaya?)Voird)bo,[(osVb%a?)Yor] FO" |CarVatrotrdl) a+18|CuVatroiea?) ad | +2[varronaVarra-v rae]. (2.20) ! k,Self-Inductance ofaRectangle | “en !t a=radius ofwire, 1=length ofrectangle, b=breadth =. V4(b+1)? 407+a Vot41240 Vo?41241£24)(041)log]——7>2"LyogJ Jes] i VTS Lon-yvicera. (2.2) | i i i i | |29 223. Concentric Solenoids (Current Sheets) [20] H N.i H A H 4% i H .ee H ' x H ! \, |Fraune23 \| lensccssescssseccccssccree! Homannantbymanaan rote tt risoftbt He by +6,% anna us008.8 NjondN,oretotalnumbersofturns 1) 20-4» FF(2)ZaratePieeaPetes} wetnla natu|pe 2?)—2eshteaeet) (2.22) ooCidaa)Te>7,/28(2st1)(2e+2) (2043) on Theseries isrelatively small compared towhen a,/2b, issmall. P,(j) isLegendre polynomial andP)(»)=d[P,(u)]/du. Ifthecoils carry currents ofstrength I,andI,(incgselectromagnetic units of current, thetorque oneither coil, tending todecrease 0,isP=-I,1,0M/20=I,I, sin62/2, or at CHD) Ear PPiggy(HIP3,(ey)PogeaHa) _oh i 2)Pager(HIP35(Hy)PagelT=4——¥,1,703,of,-2eb (2)ss ase . 2d,08272MSan Hh, 7,/Qa(2s4l) (2842)(28+3) G23) on Case a.Axes perpendicular (@=7/2, 1=0) w=0 nh) 1-n}=(Byeee Pees) <4n{2 2—. (-1)st8{—2) —___*s_ hast? 2")(2,24) reCeerect CarreDCONN) Poemaaaiasrsy J}24 on 30 Case b.Axes parallel (9=0, u=1) T=0 WyW,7a? 2C1-w3) ears Pa,(4Page(Hy) pengSaTetf2OTHD) (2)Pas(Hr)PaoesHa) (2.28) 2d, Bye r,/ 28(2841) (2842) (2843) on This may becomputed infinite terms. (See coaxial coils, eq(2.30).) 2.4, Self-Inductance ofaCylindrical Current Sheet [21] —_|—. i DY) Fioure 24 —_ 1 b-----f-----4 The sheet consists of¥complete circular turns ofthin tape without insulating space between them. Their diameter isD;the total length ofthe cylinder is1. ny? 2LAye[enti], (2.26) L where k=D/V1?+D? themodulus ofthecomplete elliptic integral Xand.Thecomplemen- tary modulus isk1=1/V1?4D?. 31 2.5, Self-Inductance ofaHelical Wire [22] .: 1OO0 0000$ i1 Df |1 We--------- f--------4 Centers ofwire on acylinder ofdiameter D Ncomplete turns; diameter ofwire isd. The length 2ofthe equivalent sheet, isthe distance from the center ofthe wire at start of first tum to center atend of last turn. ltk' b=L,t+l|log|— ‘ 2[(BE) aelog4] 1 Nerd 1 NmD\4/8 Nrd\* tnD{2¥|log{—— = ——b=(=~: y(t {[}1«(T3206(7+41)fas(Se) WZ EREL Re gtey}coon 3L ek 2pare 7 This takes account oftherelatively small axial component ofcurrent. iL,isgiven by (2,26), moduli k,k’, asin(2.26). 2.6, Bifilar Mutual Inductor [23] Primary and secondary are helical wires identical inform, the tums ofone midway between those oftheother. Two cases are %,and¥,. Myistheir mutual inductance when the second helix isdisplaced axially from the first byone-half the pitch. When the second isdisplaced 180° inazimuth from the first, but with its extremities inthe same end-plane asthe first, their mutual inductance isdesignated by¥,. The principal part ofeither isL,given by(2.26), theself-inductance ofthecurrent sheet equivalent to primary orsecondary. The moduli kand k’are the same asin(2.26). ltr’ My=b,tl [ost saelog4] 1 ‘4HD\ 4 (8 ‘yrda\? 1/K-E RR- =log(—)+—(<-1}}14+44(—— })|--(—— . alytog+3te(sV4)Lo(SEY)(-*) Rt Rtja(1-—sin . . %(%tin*)}(2.28) 32 lent, reetytifior te4k!log‘| ! j 1 ‘4H 4/8 ‘Wd \*: =nfrsoptztoe“) (Za)fstg(F2) | 2(K-E rk] 1 1 ‘tkaes [usevee(?)]}- (2.29) | | | | i | i33 \ | 2.7, Coaxial Current Sheets [24] bent w----4-----44 cit a io i1 a a Froune 26' o rot Of i sot i u——___ i i H H Total number ofturns N,and Ny ardeds((ttt) Lytly Iy-1, ist) 2M (gybathy ltt) feth)iy(nt VI, Tt,{¥(ety )+e(e-5 w(cr35w(en (2.30) Theforce ofattraction (indynes) is¥=-I,1,34/3e Prdtitals tat) (1-1 (14tl)(latly)}wefe(opttJay(eat) ye(ott) (th .. TT, w(et: tel(coF>wi(erp oeois »(2.31) where =xw! Stora) [(2)] wOdaxu' (d+SS |(22)en (2.32) This isaneven function ofx.Itsderivatives u(x) isanoddfunction ofx,vanishing with x,and given by wr(xp=2X0188pp)s|a3-02| [#828y-(rwPC,RZ|](2.33) the+sign isforxpositive, -,forxnegative. Thecomplete elliptic integrals XandFhave modulus k,where 40,0, ze 40182Morea (2.34) ‘Theincomplete integrals F(9,R‘) and£(8,k’) have thecomplementary modulus k'= Vi-m?. Their amplitude @iscomputed by x \? sin624|——"17"? whereoxe<).(ae) a, a, 4 Thebracket withfactor 40}-a} vanishes whenx=0. Ifthecoilsarealsoconcentric (c=0),the force vanishes, and ¥becomes ATWHy[(2th )-(1y-1,yj Tate bYa 8at (2.30) Another special caseisthatinwhich thesecond sheet isreplaced byasingle turnofra- dius a,coaxial with thex-axis intheplane x=c. ‘Themutual inductance between thecir- cle and sheet 1is h(e(oE)(=))=Tywaaa w¢-r (2.37) Putts {®3)*(y}2h rT (un(ong)-wr(erp SUM) (2.38) where -F_RK w(x)=4Vaya, (Se). (2.39) sothatthecircle hasmutual inductance 4,withthenearest circular turnofthesheet, and M,with the farthest. 2.8, Toroidal Current Sheets Current intape winding ofWturns circulates around thecore inplanes through its axis ofsymmetry. Permeability ofcore isy. Case a.Core ofCircular Section 74\\ ‘ 2Loxis LesotyNt (2.40): asVatat Figure 27 35 Case b,Core ofRectangular Section Ka --W ~-->4 r | oe io 4__1______ axis Figure 28 L2HNTlogOe (2.41) 2.9, Endless Return-Circuits (Self-Inductance) a.Concentric Cable (Special Case of2.46) oir metol 4and4,aremagnetic permeabilities. ‘Thecurrent goes onewayinthecentral wire and returns inthe outer shell. The self-inductance ofthe line, per unit length, is, for low-frequency ordirect current. 0» ogyHAE[_2AE0(42)1]isWe NosTrai Tay aol (2.42) atl 22)itayn 2.43 1/1242 Log(G2)if4y=0, (2.43) 1/1*2 loga2/41 forhigh frequency (see (3.1))+ 36 b.TwoParallel Wires (Nonmagnetic) 9 yi »” 9 Froune 30 G0,rb>ota, Theself-inductance perunitlength ofthelinewhentheunitcurrent isinopposite directions inthetwowires, andu=1everywhere, isgiven bytheexact formula ot 1/1=142log——. (2.44)aya, See(2.45) and(3.3), This isderived asinsection 2,eq(I)and(J). x .TwoParallel Wires ofMagnetic Material ys ees »3 >tea te Froune 31 oy 0, go8 apet eS b+, myth, oFato tog2— Wiss42 low — 1 - 27 Sette gtreregre JonQl-e,e,e) wr a[eptle,#2e,e,)e™]o ALvtataataae™ le}, (2.48) where wynl Marl==and€,=—— andy=2(8,4B;) FreesarRearseaa ‘Thepositive constants f,,fz,andyaredefined ineq(1.13). This yisthereciprocal ofcapacity (1.15). When »,=u,=1, eq(2.45) reduces to(2.44). (See also (3.3) forhigh frequency inthis circuit.) 38 4.Two Conxial Tubes (u=1) (See section 2,equations (X) and (1) ve metal, RY air oh y< metal, Froure 32 Current goes inopposite directions inthetubes. Theinductance [/1 perunit length ofthe line is a,A} 243 A ai 2a? A L/1=2log242 atos(2)-1i roa(“2)-1-(2,46) 4taga} [43-03 \@a/|atea?|ata? Nt When a,~0, the inner tube becomes asolid wire and this reduces to(2.42) with wy=H,=1. A,a} 2a? ‘ay Whena,+4,, L/1=14+log—+——|——log|—}-1]. 4)afnai|at—ai Nr a, AR 2a ‘A Whena,-4,, L/1=-Y%ytlog +——=tos(2)-1: 41"ad-a}|43-03 \ A, Whenay4, andayAy, L/I=log7°1 39 e.TwoEqual Bars ofRectangular Section [25] + : -t4 insulator9 “tT + eo----w----+4 YoYq ond8%, Since Dy,=Dy, (given byeq(M)), theformla (4)forself-inductance 1/1perunit length ofline isL/1=4 logD,,-4logD,,, which leads to 1 E/L=SSGLP(48)HF(8)VaP(428)—YPC)]40(428/3), (2.47) where asinequation (N) P(x) =-2x4 logx +(1-6x?4x*) log(1+x?)-8x(1-x?) tan“ x, (2.48) This satisfies the identical relation inx(eq (0)). Pon=xt(2)12xogx-bex(-2?). (2.49) When x<1, wefind 1 asx ¥ (-1)1x28 P(x)=2x4log=-7x?4+. SWOT eee a°B 616OYaca aI) (2-50) This isobtained byuse ofthe series =tn = an2y= -y)enXt “y= ayyntn XOLog(14x?) 2(-1)"9andxtan"?x2Com2a. Equation (2.50) might bepreferable to(2.48) forcomputation with thin flat, strips asinthe figure, where y,or5,orboth, are small, that isg,orb,orboth, are small compared tothe width w. Inthe other extreme (¢orborboth large compared tow), the expansion ofP(x) is required forx>1. This isobtained from (2.50) byuse ofthe identity (2.49) which gives for x>1 = 25. (aie P(x)=2(1-6x?) logx-7x? +=~4nx(1-x?) +:‘—_—2S °6te ox?)40x2maT RTO OD 40 ‘The derivation of(2.47) depends inpart upon eq(M), which is i 4n 25 AL090,=545P(5)-Z b-4Logut, (2.52) With thesame. function F(x), itisfound that Gray's [26] formla for410gD,, may be written ot -\ 1 25 410gD1,=555[F(7t8) YaP(#28)~'/gF(y) ]440(748)+4logw-=, (2.53) which, with (2.52), gives (2.47). The formula ofGray, corrected byRosa (27) is 1 yt402021ogD,,=—25ut0ts[ce420 2(otE22}2ogput(e2094] 1 ys 2) ytacasa2aE227)#Daogcurscenny afer(ut) giaeT minus the sane terns with »=0, aw 5tenetgy? aft2 #2((e420)? canttw? (0420)tanTH -Stan ay?ie?)9tanegy?A) 2[ce+0)ean}4,su?(240)can£2]449cana eantZ}. (2.54) Byuse oflogxy=logxtlogy, andtan“!x=1/2-tan“! 1/x, this formla maybeput inthe form (2.53). a 3.Frequency Effects [28] 3.1, Skin Effect inConcentric Cable 5metal Hiendpy a} Freune 34 metol 5 y BN oir 5 Qe cy Rbininy BE Theresistivities p,andP,areinelectromagnetic cgsunits, oneofwhich isequal to(10-%) ohm-cm (for copper 1/p~0.0006 andforiron 1/p~0.0001). Forfrequency f,letRy,andLy,denote resistance andself-inductance oftheline per centimeter length, when the current flows one way inthe central wire and returns through theouter shell. (See eq(2.42) and(2.46) forlowfrequency.) Forvery high frequency VE\P Vegery cormve (3.1) beywolott4h|?Pa]1tee tela, a,IWF, For any frequency f,the resistance and inductance may becomputed byuse ofcertain tabu- lated functions, which are the real and imaginary parts ofBessels’ (and Hankels’ function ofthe first kind), these having parameters 0and 1andargument xVT, where xisaposi- tive real. Theresistance Py,andinductance Ly,areobtained byequating real andimaginary components inthe complex equation R @,Qu, 2u, Jy(x,Mt)type] log+g — (3.2)nf a, [Yt7) 42 where _Hy 2H, Uh, x,=27a,|aX,=27a, afand-x,=27a,| af Jo(xVOLVER (xVI)I-LVTS,(2VTE) (x,7D)=Cty,(x,VE)*CYTE) (x,YI)I-LVTI, (YD) LY(x,VT] where Jo(x¥T)=uy(x)ttvy(x) andWIJ,(x¥t)=u, (x)+10,(x) ALY)(xVE)=0 5(x)+174(x)andTHE!)(xVT)=0, (x)+17,(x) ‘Theeight real functions u,,U,, U,,7,, (n=0,1) aretabulated inJahnke-Emde’s "Tables of functions," pages 246-258 (fourth edition,1945) for values ofxfrom 0to5.99. For larger values theasymptotic expansions maybeused and lead tothehigh-frequency formu- las given above. 43 3.2, Proximity Effect inParallel Wires [29] on a, Y oot. & & 6 . Froune 35 a Hip(Permesbility) p,(resistivity) b=90, Current goes one way inone ofthe wires and returns inthe other. See (2.44) and (2.45) for low frequency. Theresistance R/1andself-inductance 1/1perunit length oftheline (ofboth wires) for high frequency f,are given by a Veea 2 rro|(viaktese (yattent te-r)a"2] a, a; ay a,ji-e7 (3.3) R/L amearer where yisthereciprocal ofthecapacitance perunit length, sothat by(1.15) 1. b?-at-a2+V[b?-(a, +a)?]L0*-(a,-a,)*] 2722108]—?4 4 ofl2a,a, 2a,a =2log] —————__—___+ 2) b?~at-ai-Vib?~(a, +a,)?J[b?=(a,-a,) 7] For equal wires ofthe same material, these become 28BE ueaVb?-4a? (3.4) ruses?)RL L/i~4l(eet mtd /1~4Log 2a ‘ont 4 3.3. Single Wire Parallel tothe Earth 32 Y earth (Resistivity p,,#21) py ye.wya h (3.5) ntVh?- a?)R/L 4,Legendre Functions That Occur inthe Formulas TheLegendre polynomials P,(x) andtheir derivatives Pi(x), (where nisapositive integer orzero), occur informulas (2.4) to(2.12) andin(2.22) to(2.25). ‘These satisfy the recurrence relations (2n41)xP,C0=nP,,OOFEDPgCO), aD (2nd1) Lax? PAGdn (mt1)LP, 0)=Pagy OO. (4.2) They areeven oroddfunctions ofx,according asnisaneven oroddinteger _s s[lex\_ (stn)!P=20)(7B)en (4.3) Co ex)? (tnt)! rey<3 (12x) stmt)! Pao4d|»(5)at(et1)!(n-Ina)! G4) orinpowers ofx, eccayn ST(eex834Ktn) PaOV)2st(n=s)'PCa#k) (49) eccpyny FT(D8x28P84tn) PanesCO=C-D) “23!(nes)(34%) (40) 45, The last two equations give P,(x)=1 Py(x)=x P,(x)=14(3x7-1) Py(x)= (5x9-3x) PC 2%,(35x4-30x743) P(x) =(63x5-70x9415x) Po(x)=p(231x6 315x4+105x7-5) Py(x)="fg (429x7- 69325+315x3-35x) Py(x)="og(6435x9~ 12012x6+6930x'—1260 x?+35). (See references [5] and (11].) ‘TheLegendre functions 0,4 andP,.y occur inthecapacitance formula (1.19). The function Q,appears inthegeneral inductance formula (B)ofsection 2andistheorigin ofthe elliptic functions in(2.1) (2.16) (2.26) to(2.29), These are infinite series that occur frequently for real argument greater than 1,sometimes written cosh 6,where 8 isapositive real quantity. Var) vue coshg)=VOOM4)4twee OY)v4ent 0,-4Ceosh A=") @ PUR,v4¥h,v41; 67%) ceocluekle$7, -29pMCS4K)P8404) 4.7 °a? siP(etvel) an Py(cosh £)=P_,.y (cosh 6)=F(14, Yt, 1;1-077) hs Sy028)(34K)(styth) (a) “YaOH) ates! . or sinh?8[0,y (cosh6)P4(cosh)~0;y (cosh6)P,_y(cosh6)J=1, (4.9) where P'(z) denotes dP(z)/dz, ete. 46 ' The hypergeometric functions in(4.7) (4.8) may betransformed, leading inthe fol- lowing equivalent expressions; sh2)=~7Let) fr, BYa1p(y4th,v4) ;sech?2 2,4(coshA)vere (4sech5)FOAM,vAMg,241;seeh?Z (4,10) Py (cosh £)=(sech 6/2)! P(Y,-v, Yy—v,1; tanh? 6/2) =(sech 6/2)?"POLtv,1htv,1; tanh?2/2) a) The identity Pla, B,¥5 Z)=(1-2)7*F Ply-a,7-8,752) (4.12) shows that P,yisanevenfunctionofv. The recurrence relations (4.1) and (4.2) become . 2vcoshBP,y(cosh6)=(r4%9) P,(coshA)+(v-,) P,_,yeosh6) (4.13) 2Qvsinh? £P)_y(coshA)=(v?-Y) [P,,ylcosh A)~P,_,_(coshA)]. (4.14) Thesameformulas aresatisfied by0,y. 4Inallthese expressions vmaybereplaced byanyinteger n.Thefunctions P,y are even functions ofv,butthefunctions 0,yarenot,exceptwhenv=n.Inthatcaseeq (4.7) gives 0_,.4(z)=0,.y(z), where nisanyinteger. For the formulas given above visaninteger n,sothat itisnot necessary tocom pute these functions bythe series (4.7) or(4.8) inview ofthe many excellent tables of : elliptic functions. Ifwefind thetwofunctions forn=@ andn=1, namely, 0.andQy, t anyother 0, maybecomputed by(4.13). Similarly, ifP_yandAyareknown, therecurrence relation (4.13)givestheP,-Xfor énol. The complete elliptic integrals x(k) and &(k) with modulus k,where O0<k<1, are given by 71 .[%ae7 ime [= 2EGF, Malik =) Witaieinte (4,15) ” w .BDrigsMytsb)=f Vi1-2?sin?6a0. (4.16), Thesame functions with complementary modulus k’=V1-R? aredenoted byx’andFE, respectively. (These £’, K', etc. are not derivatives.) 47 Legendre’s relation between the four is Ke REE, (4.17) Let coshoe Rt=sech? 6/2 sothat Rit=tanh? B/2 (4,18) _2k .al=k sinhA ePa ‘The equations tobederived are O.y(cosh A)=kK. (4.19) r-k Qy(cosh )=2(£4)az. (4.20) ua .FP-x(coah A)=RK", (4.21) ZPy(cosh A)=28'~ kK’ (4.22) FPycosh A)=5) : ; Placing v=0 ineq(4.10) gives byreference to(4.15) and (4.18) 2 9.y(conh6)=0.y (Z,-1)=2aPCVys Yes1sRAVER, which proves eq(4.19). Similarly, taking v=0 in(4.11) gives FPaleosh A)=RF(Yy, Vy1,t=RE', which proves (4.21). The proof oftheremaining twoequations, (4.20) and (4.22), isnotsosimple. For this wemayplace thenotation of(4,18) ineq(B)and (B’) ofsection 2,sothat RP (x-x')24(pnp')?- 2 cosh——= 1+——ee] PMNpp aap" co (4.23) p2z__4e0" (xox)24Coro")? 48 |Equations(B)and(B‘)become(witha=¢-¢')1 - k 2sfiahaa =Dytanglcoshfeosna (4.29) 2Y1-k?cos’2n=O { i 2 72 cosnada 2cos2n8 0,-ycosh6)=0,.#7)=f =(-1*i082nd | aleosh =On.y(gat) Me[peepee [Vi-Praat oes(429) |Takingn=1in(4,25)gives 2 _,[7/2(2sin?6-1) 04(Get)8POsire”? 2ia1s(L-R?sin?) 5f? ae “a Je Wi-Risint o lo|=Wi-R?sin? 6 E-E =2(7)RE, which proves (4,20), For the remaining eq(4.22) take v=1 in(4.11). This gives Py(coshpr=Zet(33a,02)=2ralg=MyLsh!?)by(4.12) 2% 2” ght Dee” 20 fan's yee Bywriting out theseries for#’and£’itisreadily found that 2BRRP(AMy=My15k!) (4.26) Hence (1/2)Py(cosh,A)=28'/h-RE",whichiseq(4.22)tobeproved,Hencethefunction Q,-% (cosh f)andP,,(cosh 8)maybeevaluated byuseofanyofthetables referred toin section 6that give the complete elliptic integrals £and &asfunctions ofthe modulus k. This would apply toeq(1.19). Incase ofthe mtual inductance ¥between two coaxial circles, the formula (2.2) gives W/Va,a,=470y(2/k?-1), andthis istabulated against k?intable 2ofNagoaoka and Sakurai [7]. 4 Itis found that the functions fan’Fetpeo'y) ((x2x!)249-0")? =yyy(14APP andPay(142XHore") 2-24.(ae i Tee) 49 satisfy the partial differential equation (22) oo (4.27)x PptPp inthecylindrical coordinates (x,9),and also in(x',p'). The canonical expansions in various systems ofcoordinates of0,4 with this argument areobtained inreference [12]. From (4.27) itisfound that if 2 Genito)"= “Fe p! 1p ere yas’, (4.28) 0,0)vehlefleeOe(Tap then ent(ozeo3s8) (o%0,)=0. where(x,p)isoutsideSio (4.29) =4np4t(x,p), where (x,p) isinside $ which may bewritten 24ptytpMYya 5jlpated when(x,p)isoutside$ ro (4,30) =-47f(x,p) when (x,p) isinside S Forthecase n=1, U,=Ay= the¢-component ofvector potential ofacurrent distribution whose ¢-component ofcurrent density isi=f(x,). This iseq(C’) ofsection 2. For thecase n=0, U,(x,p) istheaxially symmetric potential V,ofaring distribu- tion ofcharge whose density isf(x,p) inthe ring ofsection S.Hence 2 (xox!)#4(p-p')? V(x,=zff"Fle"p04(eee Jax'dp’ (4.31) xP=Te)J,VOT 0% Top? x!dp 7 lL A v?VCD" 454DYHHO outside § (4.32) =-4f(x,p) inside $ Hence thepotential at(x,p) duetoacircular line charge ¥intheplane x!with radius p!and coaxial with the x-axis is x.(on') lore’) *) V(x,p)=—=—0.(1pGeeMore)? . xp)‘VEROy pp (4,33) 50 5.Derivation ofSome Formulas 5.1. Eccentric Spheres andCylinders (Internal) Equations (1.11) and (1.12) Equations (1.11) and(1.12) arederived byuseofbiaxial coordinates adf,de- fined bythe transformation xttystecot(=)sherem0 or ___¢sinh8 aacosh £-cos a GD __¢sina 5.2) ¥*cosh A-cos a (. eee. eazanoasTosa eeYcoshB~cosa (5.3) Vaxtyazi= ¢¥de"+ap"Trap (5.4) cosh f-cos @ Thefamily ofcircles, 6=constant, hastheequation ace eothA*4yt=(SY thgeXtutte? (5.5) xe vHawa) *°F2Oaex : ‘Theorthogonal family ofcircular arcs, a=constant, is lyytee? x*4(y-ecota(S), orcota=Xtl2" 6.6) sina 2ey ‘Thetwo-dimensional potential satifys 7)app)reeBreesoNop14092)7-0. (5.7)D3 : a tDy Fortheaxially symmetric potential, Laplace's equation with cylindrical coordinates1(or+03+42,) V=0becomes D,2403+ ~~) (pr)=0 (5.8) oP Gin? afPOO : ‘ 51 The correspondence ofthe (x,y) half-plane (y>0) andthe (a,8) strip (0<a<m), (-@<p<m) isshown bythelettering infigure 37. 2F Alo +Béo tose 22° ge®4 4 Bhsg% iva4/ \ ™5i eA Ao Barco OTOT Froure 37 Betoo© it CcB B+B,>B,-------+---—--- ee Vanea Bro “yA iOec ‘,|| BeB40 o, Be-co 'C’ c’ c’ { t, 4 “0 “4 aor Thethree constants c,f,,andf,aredetermined byeq(5.5) interms ofthegiven radii a,=c/sinh 6,,@,=c/sinh A,andthedistance betweencentersb=¢(cothf,-cothA,). Thesolution ofthese three equations forthecase ofinternal circles asinfigure 5is given in(1.10). ‘Theinner circle ,offigure 5isthedotted semicircle offigure 37. Inthecase ofcylinders thetwo-dimensional potential between these cylinders is : Ai-BW(A)=[#5] ¥,for£,<A<py. (5.9)| Thepositive charge perunit length oncylinder 2is0,>0; thenegative charge on1 isQ,,where =9,<22 i”27)q=—_1_ tre J,(es208-2)" 52 so that Q 122-¢/eu=— 1_, 7! "3B, -By) which iseq(1,12). Toderive (1,11) for eccentric spheres, one within the other, wefind the axially symmetric potential between the spheres, satisfying Laplece’s equation inthe form (5.8) for 62<A<Ay 2ewe (244) (BB) a,A)=",V -cosa. — EP) : . ¥(a,6)=", V2(coshB-cosa) aah(HTC, P,(n) (5.10) os where 4=cosa andP,(u) istheLegendre polynomial. This potential vanishes ontheinner sphere 6=,. Toshow that ithastheconstant value 7,ontheouter sphere where £=£,, the normal series = 1 rw)=>(14%)Pq(ue)fPla")Pq(u'du!for-1<p<1aa m= may beused. Since u=cosa, wefind for0<f, 1 Pa(u)du —-e~(ntk8 fPokal (5.1) 1Y2(cosh B-n)(ny) which gives the normal series 1=>=(no)6 - WoahEtre)" eP,(u)for-1<u<l. (5.12) om (Equation (4.24) istheFourier series forthissamefunction.) Taking A=, in(5.12) shows that V(a,8,)=Vy. There isapositive charge Q,on sphere No,2and«negative charge ,onNo.1,where (since yisnowreplaced bythecy- lindrical coordinate p) -lf" ar" er” sing OF -ctar dp-0,-0,=72 ["2n0,(2)a =fffcoatcow(an)20-Sf(2 70,ran72ate2Je(comb,cowa)984°"2Ja\2Af(coshA,=H) This gives byuse of(5.10) 2e-/*+4)Bp 1fiPau)dy dereDS)mime ama J,VUeoskFyn) ome 53 orby(5.11) with A=A; 2RFge(9K)(1482) ‘(antyb Sr,>& =2¢>ae, Ve ‘sinh(mt%)(Ai-F3) Joes(anti)By-Ba) oy 0 where 0<f,<A;, which proves eq(1.11). When thecircles becone coaxial 6-0, butc-, and2be~a3-a?. Hence eq(1.11) re- duces tothe coaxial case (1.5) and eq(1,12) reduces to(1.6). 5.2. Eccentric Spheres and Cylinders (External) Equations (1.14) and (1.15) Inthis case the circles are external. The circle No. 1onthe left isA=A,<0, and the derivation ismade with A,negative. Atthe end wethen replace f,by-f,, sothat in figure 37circle No. 1isA=-A;, where £,>0. This isdone tokeep thethree constants c,By, and A,all positive, asstated inthe three equations (1.13), which have been de- termined byuse of(5.5). Hence with A,negative, the potential between the cylinders is vipy=[ £22] 7,tor0>8,<A6A,<0. (5.13)Ba-By Asbefore, 1 "V 10-0-2 fFHeesgeS dniSHep,248,-P*) 20 0,/¥,=C/en=—_+_,2(B,-Ay) which becomes (1.15) onreplacing -8,by,. For the case ofspheres ayinkl 7sinh(n+%)(8-8) ¥(a,8)=",Vilcosh B=cos a)A©—_SstN48)SF“P . (a,8)=1,VilcoshB-cosa)SsiantnaieBPa (5.14) We now find nyeffar du -0,=0=f()—“K. VENT Neb]p,(eosh 8-H) or 2 ana, , osew ifPau)dy c=;—_—_*__— y)————, 7oedDwink(44)(B.-A)°"*21VBCcoshBw) or 54 Since f,ishere negative, wemust write eq(5.11) “Palw dautnstila,|_— olnekla (gf eee is *forB,<0, 2L:Vacosh =n)” ° 7Bs so that xe eeayre,)i” eclny sinh(n+)(4,~8,) .Leeintl” or o where 7=2(8,-A,) when ,<O and£,>0. Onreversing thesignoff,,thisgiveseq(1.14). Forthelimiting cases8,~O,in whichthesphere orcylinder ontheleft offigure 6orfigure 37hasaninfinite radius, wemayplace b=a,+h anda,=a,, Whena-of c+Vh?-q?, B,=1og(h+Vh?~a?)/a, s0eq (1,14) and(1,15) become (1.8) and(1.9), respectively, ‘Thepotential between cylinder and plane f=0 (fig. 4)is ntVhP~a? vip)=4 v,for0<p<p,=logtte ==. Bya . Between thesphere andplane thepotential iseq(5.14) with 6-0. 5.3. Derivation ofEquations (1.17) and(1.16) forOblate Spheroid and Circular Disk With oblate spheroidal coordinates (a,8) the(x,o) half-plane isrepresented onthe _.(0<asm (a,A)strip(032) by xtipsic sin(at if) where c>0 or x=-c cos asinh fand p=c sin acosh § s0 r=Wx?tp?=cVain? ‘Btsin? a x? p? . HFsinh?B'GFqoontBo}(confocal ellipses) ox? 2+—£—=1(confocal hyperbolas). ¢?cos? ac?sin?a ~ 55 The equation for the axially symmetric potential (2200+) (An 4sin? a4cosh? 6 has solutions 7=Q,(1sinh8)P,(cos a).Theoblatespheroid 4-8,hassemiaxes aandb<a, where c=Va?-b?, sinh £,=b/c andcosh £,=/c inn py=btog! 2thBH1(ge, 10,(tsinhary10gSahB=peisit (sechA), 80 20,(t sinh B,)=sin™! (c/a). ‘Thepotential outside theconducting spheroid A,atpotential 7,with charge ¥,is(for B,<B<@) 40,(1sinhA)sin"!(sech (A) ¥(B)20-——— —" 5.15)VGC sinhA) gin(e/a) (5.15) F i. ony andw=Limit(r¥(6)]= Litesinh61(8)]= SG" which iseq(1.17). 5.4, Derivation ofEquation (1.18) for Prolate Spheroid With prolate spheroidal coordinates the (x,) half-plane isrepresented onthe (a,A)strip(O<acr), (0<p<m) by x+lp=-c cos(attf), where c>0, or x=-@ cos acosh fand p=c sin asinh f,so r=Vxp?= cVainh®B+ costa. Hence 2 2 —~__ +—*_-=1(ellipses)c?cosh? 6c?sinh? 8 a 2—_-_£ =1(hyperbolas).c? costa c?sin?a 56 ‘The equation for the axially synnetric potential 1 (:8+08+ tar) en? 4sin? a4sinh? has solutions ¥=0,(cosh A)P,(cos a). For theprolate spheroid =A, with seniaxes aand b<a c=VOID? and cosh A:=0/c, “acoshBt1 BL(wert) 2glcoahA)=Y,ogeeEttLogcothie=log (+2), Qg(cosh A,)=log (ate) /d. The potential outside the conducting spheroid ,atpotential 7,with charge 2)is (for fy<B<e) Qe(coshA)_logeoth6/2 V(B)=7, , (5.16)(P=V10coshBi)"Toglare) 7B and Hy-Lin(rv(p)]=Lin leVotah?Brees?oF(A) et “Tog(arc)78" which iseq(1.18). 5.5. Derivation ofEquation (1.19) for aToreid The strip (-7<a<m), (0<f<@) ofthe toroidal (or "ring") coordinates, represents the (x,) half-plane, ifthis iscut from zero tocalong the p-axis. The equation X+tp=-c cot(at18)/2gives _cesina __csinh8 Gosh Peonan"! "cosh B-cosa (5.17) wk ynpeora reVxeeneeVWcoshB=cosa Vactap Vaderayaa setah (5.18) cosh F-cos a 57 The family ofcircles A=constant, each member ofwhich generates atoroidal surface byrotation around the x-axis, belongs tothe equation a hpy2=—o. x44(p-ceoth A)*=TG. (5.19) The equation ofthe family ofcircular arcs, orthogonal tothese circles, is a (xtecota) 4pia, (5.20)ain? a ‘The equation for the axially symmetric potential (omc )(p47)=0 5.21) * ‘4sinh? 6 has solutions of the form ¥=¥2(cosh F-cos a)(AcosnatBsinna)(CP,y(cosh£)+D0q.y(cosh f)). ‘The third ofeq(5.17) shows that spatial infinity, (r=©) corresponds tothe point a=f=0. The first two ofthese equations show that A=+® corresponds tox=0 and p=c= the radius of the focal circle. Ifthegenerating circle offigure 7hastheequation f=, itisevident from (5.19) that c=Y4?-a?andcoshBt. (5.22) Ifthe toroidal surface has aconstant potential 7,and charge #1, the Newtonian potential atoutside points where 0<f<f, is a.At!VSRERSSOn-(coshf;) ¥(a,P)==! VBcoshFeces@)<PycoahBy)P*-Mlcoshf)cosna, (5.23)=o where €9=%, €,=1 forn40. This vanishes atr= (i.e., when a=f=0). Onthe surface A,itbecomes _ ¥(a,y)=V, VBeakFy=cosa)—>€nn-¥(cosh B,)cosna 0 =V\=constant byeq(4,24). 58 |ByafundamentalpropertyofNewtonianpotentials,thecharge¥,onthetorusis emi . cosh Freesa HL = te F(a,-limie(rV)=Limit ¢Vfeer A) Aer, Sy On-¥(cosh 21) , ADinit >‘npacleask aryaHleoshAeosAa. =o Since P,_y(1)=1, this gives Wyde>,DnewlcoshBy) v7 "P,.ukcosh By)" m0 which iseq(1.19), since c=YA?~a? andcosh6,=4/a. Theevaluation ofthese func- tions byelliptic integrals isdiscussed insection 4. 5.6. Self-Inductance ofaSingle Turn ofWire Equation (2.15) With cylindrical coordinates (x,)thevectorpotential4,=A(x,o) atanypoint (x,p) inspace isbyeq(C) ofsection 2. Hy 4 2 (5.24)= Keep! as! 7 pAACx,p)afftp004(1452-) S where D?=(x-x')#4(p-p')?, andtheintegration istaken with respect to(x‘,o’) over theupper circle ofradius ainfigure 18. Also byeq(D)ofsection 2theself-inductance isgiven by beanffoe(o)-pha(x,p)as (5.25) integrated with respect to(x,p) over thesame circular section. Since (x,p) and(x',p') areboth points inthis circle, wemayuse theexpansion (ose)li Dn2pp' tt eC pt 1 1 a >Sept)at atLegit Yatntv84Vpn)24(841)|.(5.26) = This isvalid ifD*/4op'<1, which will betrue forallpositions ofthepoints P(x,p) andP'(x',p') bothwithin thecircle, provided that a<4/2, which will betruehere, 59 since itisassumed that a/A issosmall that terms smaller than a?/84? log84/amaybe neglected incomparison with 1. Hence for n=l, eq(5.26) gives, tothis approximation, Dt 3(_p? D? Dt 2(%)-[ 3(eh.—F-$4(1-log2)+¥,(1-6log2) 5.27 O(+5257) =a(gop) Jlowzgrte(i=tow2)+%4(1-6og2)77 (5.27) Let y=p-A andy'=p'-4, sothat xandyare rectangular coordinates with origin at the center ofthe circle. Then, tothe second order ina/A, .pt\_| D_yty'yity'?o( 4)] 204(1+32) =[442togYsgattag(1431085, (5.28) For the assumed current density __1 eye tsap) , (5.29) the total current is1,which gives pfLcd4) yt ver( ar SEY Ley Cae (5.30) Then as(“4as[y4] Ki(p=AL(14)"-4 You eile) al ty matPLtigtCage (5.31) where ,=b+¥4 andC,=%,(»-1) (5.32) 1=d+%y 2="{d*-7). i. ‘This gives % ' Py 20°10" )O4(145 Jaap CrgHage||At?lows, (HY PeMWaye,weZ( 2| (75CrHMO,Fetags(143loge7) (5.33) Touse this expression inthe integral (5.24) itisbetter touse polar coordinates, plac- ing x=rcos @and y=rsin@, sothat Dt=r2-2rr! cos(9-0')4+r'?, (5.34) and 60 aye Y(t Vcosn( 8-8") : = 2 . 0-06" “logr'= ¥(5)coats8)isign, (5.35) T The result ofintegrating eq(5.24) is us PAC, p=ASE9IHF (780474 (78) (5.36) where atc, BA)?ratr)=3-2(1s22) 08(5 (5.37) C.r*]rsing rcrsay=-[ase,- eft? (5.38) ~2Tg 3, 841,(7.0)=Z5[2-20, +70,-Hoe%] 7316-3togacaeinteJeenl? into] ranErBloge+(4-20,ain?o[+|540,140,ainto(5.39) With this, the integral (5.25) gives tnd (\a] aA)7,a? *)22,7 ay{Ls40+agroe(84)teat2e(ttg)-zSamy (5.40) Finally, multiplying bythe factor, 1 a a?haygy(p-1)Bia1-0,4%, 5.41) FLD O-Veasl-Coagae C where Cy=2b(b-1), gives 7 3\a? eA)7.a?[7 (u2)-2 s-ara{[14(40,-24 8)25oe(M4)eg[Leen120,4p]-Fea|e(5-42 Onsubstituting theexpressions given above forCy,Cy, and0,, itisfound that am, aBA)7(ou)(o72/3)(2"} 4a{[pca0sn 2jioe(%4)-f Te4)}(5.43) which iseq(2.15). a Exact expressions forthemagnetic field andinductance ofanytoroid with this cur- rent distribution maybefound asnormal series ofring functions, using thetoroidal co- ordinates ofreference [20]. Itisthusfoundthatforb=-5/2 theexternal magneticfield isthesame asifthetotal current were concentrated inthefocal circle. ‘This is true forthemore general case 1,=Co-* ?/(8). Togetanexpression forthepotential A(x,p) when thepoint P(x,p) isoutside the cirele offigure 18,theapproximation (5.27) based on(5.26) cannot beused unless the distance ofPfrom thecenter Cissmall compared toA.When this distance isoftheor- derofmagnitude ofAorgreater, while P(x‘p') remains inthecircle, itissufficient touseTaylor's series, with(x,o) fixed andthevariables x‘/Aandy'/A(=(p'-4)/A) small. For brevity, let mx) 24(p=p')*_ (xox!) P4p24p!® garg SRE ore2Gon!) ttete2p 2pp ‘axto?tA?0 ede 80-f, when x'~y'~0. Then, tothe second order MgC4)Dy(Ey)H(x'OUD, #4(x*D,,4Y'20,, H2x'Y' Oy), (5.44) whereQxxisthevalueofDE+y(#) when€=€4(x'=y'=0) andsimilarly, Q,,and0,47D,0,04=D,D Oy. Fromeq(4.27), withn=1andvariables x‘,p’,wefindanexactexpression whenx‘=0 and p'=A. =.DaxtOyy= Fa0K(Ey) (5.45) Byuseof(5.44) with thecurrent in(5.31) ineq(5.24), itisfound that -o)/4 a? a? sexor-aV/A{frsoey Ze]ostesrga ovryresa,}, (5.46) where A nV, fo 240y=-2(40-4)oceer=- FAA|[ote#0)-2-u6e4)1, (5.47) and apkttlana)?fee. (5.48) 62 Equation (5.46) isvalid when ¢,-1 isnotsmall. Hence there remains agap, nothere con- sidered, between the ranges ofvalidity ofthe two equations (5.46) and (5.36), which could only bebridged byanequation more complicated than either. Applications of(5.46) that would require theretention ofthe second-order terms areexceedingly rare. Itis generally sufficient toconsider the total current concentrated inafilament with trace atcenter of the circular section of the wire. 5.7. Derivation ofEquation (2.16) forSelf-Inductance ofaSingle Turn ofWire Near aMagnetic Medium Referring tofigure 19let4q(x,) denote the value atany point P(x,p) inspace due toany axially symmetric distribution ofcurrents when =1everywhere. These currents are all tothe left ofthe boundary plane x=x9. Similarly, let43(x,p) denote thepotential atanypoint totheleft ofx=x» that would beproduced (with »=1 everywhere) byafictitious distribution ofcurrents that is theimageoftheexisting distribution byreflection intheplanex=xq. Then the potential A(x,p) due tothe actual currents inthe presence ofthemagnetic material with #1, where x4<x, isintheair, where ~<x<x,, | A(x,0)=4o(x, 0)4AA(x,0), (5.49)Hel and inthematerial, where xgex<+©, 2 40x, p)= Ag(x,p). (5.50)ut Bythisdefinition ofAg(x,p) and4,(x,p) itisevident thatattheplanex=x», Ag=Ay, and Dylp=—DyAg identically inp. Consequently, 4iscontinuous, which makes B,continuous. Also thecontinuity ofB, isassured bythat ofD,A/u, The inductance ofthe turn ofwire near the material asfigure 19isby(5.49) eel ss a . bet,+47 Jfteerecxer Ss (5.51) integrated over acircular section ofthe wire. This integration could beeffected for theformofcurrent ineq(5.31)byuseofeq(5.46), assuming that2x»,isnotsmallcom- |pared toA. Formula (2.16) assumes that thefictitious current producing 4,(x,p) isa filament ofradiusacoaxial withthex-axisintheplanex=2x,. For this approximation, weplace in(5.51) 4xdt(A-a)? 2 2(K-B) ] =249} Axor(Ana)? 9404,(2-1)-24[2D PA(X, p)=240Yy(u3a 240%,(412a[7RE, (5.52) where 44apratt . at(ava)? 5.53) 63 SincefftdS=1,thisgivesPED - Ietah,+otam[2-8)ay]: (5.54) ars ® where L,,, isgiven by5.43). 5.8. Derivation ofEquations (2.40) and (2.41) for the Self- Inductance ofToroidal Current Sheets (Tape Winding) With ideal tape windings the current circulates asindicated bythe arrow infigure 21. There isnoexternal field and theinternal field oftheunit current is#y=2H/p, where ¥isthe number ofturns, and pisthe distance ofapoint from the axis ofrevolu- tion. ‘The inductance Lisequal totwice the integral defining total electrokinetic en- ergy f. 11/80 Sfitay integrated over all space. Hence ueffweaveaur ffs 4 Pe integrated over the axial section, For circular and rectangular axial sections shown in figure 27and 28, this results ineq(2.40) and (2.41), respectively. 5.9. Derivation ofEquation (2.45) forSelf-Inductance per Unit Length ofTwo Parallel Wires ofMagnetic Material Referring tofigure 31, the current +1flows upward perpendicular topaper with uni- form current density i,=1/7a,? incylinder No. 1.Thecurrent density incylinder No. 2 isty=-W/nad. The only components ofcurrent density and ofvector potential are the z-components where the z-axis isupward perpendicular tothe paper. The general field equations B=uH=curl Aand curl #=471 give B,=D,A, By=~DyA, B,=0, where 4(x,Y)=A,. Hence 4u (0,240,?)4=-— incylinder 1 a,? Aus (5.55) =t— incylinder 2 a,8 =0 in the air between them. 64 ‘The boundary conditions atthe surface ofeach wire are: 4iscontinuous (continuity ofnormal component of8), (5.56) 124, continuous (continuity oftangential #). (5.57)Zn With plane polar coordinates (r,,6,) with origin atcenter 0,ofwire No. 1. 1 1 (0,402) 422d,(riayers, Aane 1 Pea1 r? Similarly, with poler coordinates (r,,0,) with center atO, 1 1 (0,24D?)A=—D, (729ary, A.aac) 2 2%2 rt Hence let ai4=0-n,(— )inwireNo.1 7 a5 rt (5.58) =U+u,(— )inwire No. 2 cH =0 in the air Then (D,7+D,?)0=0 everywhere. (5.59) Atry=a,, 0,=0 dD,y==D,T-= (5.60) oHUysandDyMy=DeDera x Atry=a, Dy=UtyandDyTp=—Dy,Uy4— (5.61) o=U the eMart a z where J,means outside, U,inside thewire. 65 The self-inductance ofthe line per centimeter length 1Wena ffaes,-, ffAds,ma,” ma,2 1 1 1 Mylaytna)t— |}Ud8\-— Uas,. (5.62)7a, ma, The biaxial coordinates a,f, see reference [16], are suitable for constructing the harmonic function J(a,8) that satisfies the four boundary conditions in(5.56) and (5.57). Wefollow the procedure adopted inderiving the potential ¥ineq(5.14), that is, wetake asthe equation ofcircle No. 1offigure 37theequation A=B,, where f,<0. Intheend result wechange thesign of8,tomake all theconstants c,A,, and8,positive, asgiven inthe three eq(1.13). By(5.4) the surface element dSfor integrating over acircular area bound bythe circle 8= constant is . *dad;as=2-408 _ (cosh6-cosa)? where da, d6>0. Nowsinh 6,=-c/a, andsinh 6,=c/a,. Hence after anexpression for U(a,8) isfound, eq(5.62) becomes MytHyy 2sinh? A,(x0 pw a,A)da me J,(cosh f-cos a)? 2sinh?£,¢*° 7 a.aTf apf—ZieA)da (5.63)7 da, (cosh f-cos a)? U(a,A) will befound asaseries incos na, sothe following integrals will bere- quired. a “nt coth ° A[coe nada ___oM(tcoth2)_4neDyslstn)etm (5.64) 7J,(coshx-cos a)? sinh?x ot if O<x. From this wefind, when 0<f, **@-2"*(ntcoth x) sinh?SecotXldx=e72"8, . 2waJsikh axe (5.65) The function U(a,A) that satisfies the four boundary conditions in(5,60) and (5.61) is 66 Inwire No. 1,where-™<A<f,<0: Wa,B)=H,-C 5-28, >Ae"F-81)cosna. (5.66) Inwire No. 2,where 0<f,<f<}@: D(a,P)=—py-Co- 2Ay+>Bye-*'8-42) cosna. (5.67) ‘1 Intheairbetween them, where £,<f<A,: 7 $7[Ansioh(A,~6)4Bysinhn(5-85)]cos D(a,6)=-C426+>[SeshateecehBatokntBAs)osna,(5.68)on where” 57A,sinhn6,-B,sinhnA, C=Ansint anny. (5.69) .°easinhn(A,-B,) This makes J=4 vanish atspatial infinity (a=6=0). The boundary conditions require 2U+e,)“1 mop __2C1tes) ny)9288). 1% : Aneeeroererera seu V)e2"B1-(lte,)e-"7] (5.70) - -2(1+e,) - 1By ="7)e2985-(14.6-"7)], . Bye -™ArWwe 76777) (ltere'y Yen2"82—(1¢.e-"7) ] (5.71) where alae aed =2(B,~ eT CrateTtnd728,72). (5.72) Performing theintegrations in(5.63) byuseof(5.68) and(5.69) gives B/em=471249 8,-f,)4 >(Ane"81-Bye-82 7 sEEAI2(A)-8,) = 1 '. (5.73) 2SBasegereL141te0-08 m= +(1beg)(Ite e777 0-28 ~2(1t61) te,)e-*7] 67 Toobtain positive constants for computing wenext reverse thesign ofA;, sothat y=2(B,+62), a8ineq(1.13), where 6,,6, andcare allpositive. After this change we find that when 4,=,=1 the formula reduces totheknown correct expression, say [g,that isgiven in(2.44), where a 1 bg-1=2log——=2( B+Bg42>=(e72614e-2"82-26-"7), (5.74) aya,7 = Subtracting this from the expression for L(with positive f,) gives the eq(2.45). 6. References [1] E.B.Rosa andF.W.Grover, Formlas andtables forthecalculation ofmitual andself-inductance, BS Sci. Pap. 169, revised 3ded(1948). [2] F.W.Grover, Additions totheformulas forthecalculation ofmtual andself-inductance, BSSci. Pap. 320,537-570(1918);also,Inductancecalculations (D.VanNostrandCo.,NewYork,N.Y.,1946). [3] J.H,Dellinger, L.E,Whittemore, andR,S.Oulds, Radio instruments endmeasurements, NBS Cir. 74, 2ded,235-241 (March 1924);. 235-241 forcapacitance, 242-282 forinductance. [4] G.W.andR.M.Spenceley, Smithsonian elliptic functions tables (Washington, D.C.1947). [5] E.P.Adams, Smithsonian mathenatical formilae andtables ofelliptic functions, Publication 2672, 260-309 (Smithsonian Institution, Washington, D.C.1939), [6] H.Nagoaka andS,Sakurai, Table No,1,Tables oftheta-functions, elliptic integrals Kand&and associated coefficients, Sei. Pap. Inst. Phys. Chem. Research, (Konngome, Hongo, Tokyo 1922), [1]H.Negoaka andS.Sakurai, Table No.2,Tables forfacilitating thecalculation ofself-induetance of circular coil andofthe mtual inductance ofcoaxial circular currents, Sci. Pap. Inst., Phys. Chem. Research, (Komagone, Hongo, Tokyo 1927). (8] H.Negoaka, J.Coll, Sei. 27,18-33 (Tokyo 1909). [9] E.Jahnke undF,Ende, Funktionentafeln, 114-172 (B.G.Teubner, Leipzig, 1933). [10] B.0.Peirce, Ashort table ofintegrals, 118-119 (Ginn &Co., NewYork, N.Y.1899), [11] W.Magnus endF.Obethettinger, Formas andtheorens forthespecial functions ofmathematical physics (Chelsea Publishing Co., NewYor, N.Y.,1949). [22] C.Snow, Thehypergeometric andLegendre functions with applications tointegral equations ofpotential theory, NBS Math. Tables MT1S (1942) revised asNBS Applied Math. Series 19(1952). [13] ©Snow, Astandard ofsmall capacitance, J.Research NBS 42(March 1949) RPI9T0, Thetwo-dimensional case isbased onthetransformation with theta-function, p.297, eq(37). Thepreceding cases (1.1), (1.2) arebased upon amore general transformation with theta-functions inwhich theclearance isnot neglected asitisinfigure 2,Experinental methods ofevaluating edge corrections aredescribed by ALH,ScottandH.L.Gurtis,J.ResearchNBS22,747(1939)RP1217. [14] C.Snow, Potential problems andcapacitance for«conductor bounded bytwointersecting spheres, J. Research NBS 43,377 (Oct. 1949) RP2032, Thepotential field isfound infinite terms when w=nn/m, where misanypositive integer, butniseither1,2,3,or4,andthecasen=3and4 involve elliptic funtions. Thecapacitance isalso found for«conductor consisting oftwounequal spheres inextemal contact. (AS] J.C.Maxwell, Electricity andmagnetism 11,328(1692). (16)H.L.Gurtis andC.M.Sparks, Formlas, tables andcurves forcomputing themitual inductance oftwo coaxial circles, BSSci. Pap. 19,541-576 (1923-24); also see, sec. 2,eq(F)andsec. IV. [17] J.C.Maxwell, Electricity andmagnetism 11,335(1892), [18] c.Snow, BSJ.Research 3,255(1929) RP94. 68 |{19}E.B.RosaandF.W.Grover,Formasandtablesforthecalculationofmitualendself-inductance,BSSei. Pap. 169, 155, revised 3ded(1948) [20] C.Snow, J.ResearchNBS22,607(1939)RP1208, [21]C.Snow,BSJ.Research8,419(1932)RP479. [22]C.Snow,BSSei.Pap.537,24,431(1926), [23] C.Snow, J.ResearchNBS24,597(1940)RPI302. [24] C.Snow, J.Research NBS 22,239 (1939) RPLI78, [25]Francis B.Silsbee, Astudyoftheinductance offour-terminal resistance stendards, BSSci.Pap,281, 375-422, (July 1916), Theinductance ofparallel wires, tubes, andunequal flatstripe iscomputed and measured. [26]A.Gray, Absolute measurenents inelectricity andmagnetism II,partI,p.268-306 (MacMillan Co, New York, N.Y., 1893), [27]E.B.Rosa, Gnthegeometric meandistances ofrectangular areas andthecalculation ofself-induct~ ance. Bul. BS3,6,eq(8) (1907), [26]JohnR.Carson ondJ.J.Gelbert, Transmission characteristics ofthesubmarine cable, J.Franklin Inst, 188, 705-735 (Dec. 1921). [29]JohnR.Carson, Wavepropagation overparallel wires. Theproximity effect. Phil.Mag.x11(April 1921), WAsuINGTON, October 24, 1952. 69 49.§covenmenr paneraONCE:19840=sn