G Smith Tech Rpt 624 proximity effect

Idies of Novel Researcb Centrist 1OO114-17.A-121I155 N1371-11 mlcrowave Physics Breach Ballittle Resuarch Laberterlen U.. Army Aberdese Previug freled THE PROXIMITY EFFECT IN SYSTEMS OF PARALLEL CONDUCTORS AND ELECTRICALLY SMALL MULTITURN LOOP ANTENNAS By SIn Smith ,..br 1,71 D D C Tooks lea rt ft. 124 jji 22 191T2 byproducod by NATIONAL TECHNICAL INFORMATION SERVICESpiinogf ld, Ve 22111 This document has been approved loT public release and sale, its distribution Is uakin rted ilvilln of nlliserlng sod Applied l les hrvard Imlversity 9 Canbridge, Uumsashmsetts wi Il I' i %% fi s IIonI Ii... . . DOCUMENT CONTROL DATA • R & D At-% witi 1 lot Wivo'.tiuV o t!litl h, wi) tol obstirl madJ Indi**tow antI*snnlnvle niumf hr enteluved whi-t1, Ihlv vDve ll t'Cltt I, e l, AIIIfl i 0 6-I1404 TING AC IivIT YCoporPa.t authorA 8.. MIPI'OlT OLCU4l f Y CLA U11W Ic A I ION Division of Engineering and Applied Physics Harvard University lb, CO 7... Cambridge, Massachusetts 02138 THE PROXIMITY EFFECT IN SYSTEMS OF PARALLEL CONDUCTORS AND ELECTRICALLY SMALL MULTITURN LOOP ANTENNAS f SRicNP 11% 9 NOE (Ty-pe oflto-port andI ricludtvU dorrig -w1#40miiOrba /rtl home, middle soft lie , #aso# home) Glenn Smith R EPORTt OATL 78, TOTAL NO. O PAGSib O 1 4F December 1971 116 38 Ba "CON C ON GRANT NO On. ONIGINA TON'S NRP'ON r NIJMOILISI N00014-67-A-0298-0005t, PNOjacy 140. 624 lib. OT [N RMIPONT NOIf (Any ohet her ',wnbera that mrmy be sdihlgnedth[i t*port) d, I r OISTAIUUTION STATEMENT This document has been approved for public release and sale; its distribution is unlimited. S --P'PLEMENTARY NOT .. 12. SPONO. NO ,MILIT AMY ACTIVITY' "Joint Services Electronics Program through (Adm. Service -Office of Naval Research, Air Force Office of Scientifici-711;l ¢ S, & L~_ A_ y r Il E ect. .o.m nia.no., . In this report losses in sjstems of parallel sound conductors are studied. Both the normal skin effect loss and the additional loss due to the close proximity of adjacent conductors are considered. The results obtained for the parallel conductors are used to evaluate the radiation efficiency of electrically small multi- turn loop antennas. DD E ,oA, 4 (PAGE 1 ) .... .-.,,• D t./' (A 1 Unclas sified S'!, 0101.607.-601 .... , ; T i-...... Office of Naval Research Contract N00014-67-A-0298-0005 NR-371-016 Microwave Physics Branch 2= BalListic Research Laboratories U. S. Army Aberdeen Proving Ground THE PROXIMITY EFFECT N SYSTEMS OF PARALLEL CONDUCTORS AND ELECTRICALLY SMALL MULTITURN LOOP ANTENNAS By Glenn Smith Technical Report No. 624 Thsdocument has been approved for- public ]release and sale; its distribution is unlimited_ December 1971 The research reported in this document was made possible through support extended the Division of Engineering and Applied Physics, Harvard University by the U. S. Army Research Office, the U. S. Air Force Office of Scientific Research and the U. S. Office of Naval Research under the Joint Services Electronics Program by Contracts N00014-67-A-0298-0006, 0005, and 0008. Division of Engineering and Applied Physics Harvard University -Cambridge, Massachusetts THE PROXIMITY EFFECT IN SYSTEMS OF PARALLEL CONDUCTORS AND ELECTRICALLY SMALL MULTITURN LOOP ANTENNAS By Glenn Smith Division of Engineering and Applied Physics Harvard University Cambridge, Massachusetts ABSTRACT In this report losses in systems of parallel round conductors are studied. Both the normal skin effect loss and the additional loss due to the close proximity of adjacent conductors are considered. The results obtained for the parallel conductors are used to evaluate the radiation efficiency of electrically small multiturn loop antennas. SECTION I ANALYSIS OF SYSTEMS OF PARALLEL ROUND CONDUCTORS 1. Introduction In a system of parallel conductors the distribution of current over the conductor crcss section is determined by two effects--the normal skin effect and a proximity effect. Both are the result of the same phenomenon, eddy currents in the conduct6rs. The former is usually considered to be the result of the net current in a single conductor while the later is due to the currents in neighboring conductors. For close conductor spacings, the distribution of current due to the proximity effect can cause an increase in the ohmic resistance which is larger than the skin effect resistance alone, i. e. larger than the ohmic resistance of the isolated conductors. The skin effect in round conductors is discussed in most texts on electroma rtic theory [1], [2], [3]. The proximity effect has received much less attention. Most of the theoretical and experimental works on the proximity effect deal with two wire systems where the wires carry equal currents in oppcsite directions. For examples, see the work of Kennelly [41, [5], Carson [61, and Dwight [7], [8]. This geometry has a direct application in the problem of wave propagation along parallel wire transmissioni lines. The only investigations of the proximity effect in systems with more than two conductors appear to be those done in conjunction with determining ohmic resistance and Q of inductance coils. Of the theoretical treatments, Butterworth's discussion of the alternating current resistance of cylindrical conductors and solenoidal coils is the most thorough [9], [101, [111. His work is considered the standard theoretical approach and is summarized in several places (12], [131, [141. The experimental work of Medhurst, however, indicates that Butterworth' s calculations of the radio frequency resistance of coils are not valid over as large a range of parameters as expected; for certain dimensions, errors as large as 190% were observed [15]. In the remainder of this chapter, systems composed of various numbers of in-line, parallel conductors are analyzed. All the conductors have the same circular cross section and carry eqtal currents in the saime direction. Only the high frequency case where the currents are confined to a thin layer near the surface oi the wires is considered. This report is an extension of the investigation of the two turn loop antenna reported in 1161. 2. The Nature of the Current Distributions in a System of Parallel Conductors A. Proximity and Skin Effects In the system of pArarllel conductors illustrated in Fig. 1-1 there are two factors which determine the distribution of current over the cross section. The first is the normal skin effect which, for high frequencies, causes a concentration of the current near the outer surfaces of the condiuctor. This is depicted in Fig. 1-Za for a single, isolated, ti c t I - SECTO CRNT I i S M I~i FIG. 1-1 PARALLEL WIRES OF CIRCULAR CROSS- SECTION CARRYING EQUAL CURRENTS IN THE SAME DIRECTION. (. z L0 00 0 3 1- a 1LZ 90, U)4 w L OU) < _ 3- -5- round conductor. Secondly, there is an additional redistribution of the current due to the proximity effect. This is caused by the magnetic field present at any one conductor due to the currents in the other conductors of the system. The proximity effect for two parallel, round conductors carrying equal currents in the same direction is illustrated in Fig. l-2b. In the two conductors, the proximity effect forces the current to the outside edges, much as the skin effect forces the current to the outside surface of the single conductor. B. High Frequency Approximation for the Current Distribution and Resistance At sufficiently high frequencies the skin depth ds for a good conductor is a small quantity compared to the cross sectional dimensions and most of the current in the conductor is confined to a thin layer near the surface. The magnetic field external to the conductor is approximately the same as the field of a perfect conductor of the same shape carrying an equivalent surface current. An expression for the time average power loss per unit surface area of the good conductor, in terms of the component of the magnetic field Bt tangent to the surface of the perfect conductor, is I~ Rs IBtI 2~R I B .i 2- 0 Watts/(moter)Z ll In terms of the surface current Ks on the perfect conductor P R ' IK5 12 Watts/(meter) (1-2) [ -6- where R8 is the suriace resistance. R _d (I -3a) a cd d(l-3b) If te conductor is cylindrical and K is an axial current density, the 5 power loss per unit length of the conductor is p I f KsJ2dw Watts/meter (1-4) where the iniegral is over the periphery of the conductor. For the isolated, circular, cylindrical conductor of radius a carrying total current I, rotational symmetry applies. Equation (1-4) reduces to the familiar "Rayleigh formula" for the high frequency resistance per unit length of a circular conductor p 4 1112 Watts/meter (l-Sa) R -0 Ohms/meter (I -Sb) Rayleigh ira = 2iia 2o which is valid for a/d >> 1 (1-6) s With more than one conductor present the current distribution and external fields for each conductor are no longer rotationally symmetric; therefore, equations (I-5) no longer apply. Further investigation -7- is necessary to determine conditions like (1-6) which insure that the high frequency approximation expressed in (1-4) is valid. Consider a system of long, in-line, parallel conductors carrying equal currents in the same direction (Fig. 1-1) with parameters such that a <<h , 0a<< 1 (1-7) 2: 2 2 n c << h0n c << I c >a (1-8) Neglecting displacement currents as compared to conduction currents, the axial component of the volume current density Jmz interior to the I th cylinder must satisfy the following partial differential equation in cylindrical coordinates (r, 0, z). 82883 r a j amm 2 r (r ) + + i0r = 0r Or 2 mz(1-9) An e time dependence is used. The solution to (1-9), obtained by the method of separation of variables, which has the desired symmetry and remains finite at the origin is (r, = (- l)Cm M (1 -I-) e dScos(PO)Jmz~r E M P d) P=O (1-10) where Mp and 0p are the modulus and phase of the Kelvin functions (berp + i bei p) [17, p. 379]. The Cp are functions of z only. The total -a- current at a given cross section of the conductor is Irm(z); therefore Ire(z} =f f Jm (r,O,z) rdOdr CmM1/t s)e " 1(z) =r=O O= -w i4 i 1%' r=00=-W(1-11) and eia) -m m/#d I(---- (I-12) mO -lirad a S The volume density of current extrapolated to the surface of the conductor is I ) M (42,-) -it eo() /-) -- Jmz(a,O,z) d- e 8 S 42"'iad M (, S [I+ a' cos (pO)I (1-13)p1 With (I-12) and (1-13) substituted in (1-10), the current density becomes I (Z) Mo( A-A) -i[ () (4 ) -o (R -)] Jrnz(r, 0, z) as= e s s S COD M (F -Mp( -f-) 1 + a' s cos (pO) p.-) p(,/2- -ie o (,,i,,--),- e(z a + o/' l(1-14) e (I-14)a -9- Where a = a -r is the radial distance into the conductor from the scwrfce. la the present analysis, the coefficients a, are assumed to be ccmplex MP numbers. When the current distribution at the surface of the conductor is sufficiently smooth, a finite number q of the Fourier series terms in equation (1-14) are adequate to approximate the current density. If, in addition, the frequency and conductivity are high (a/ds >> 1, s << a) the large argument asymptotic formulas for Mp and 0p apply ['1. In- serting these into equation (i-14) yields SI -tz) 0 -- qe m(Z) -- 1+ a' Cos (PO) -t/ -= I p__ 2 * ~~~-i [ (-()e ) -Ij d8 -1 (1-15) which simplifies to Csq Jmz(r, 0z) I z)[ + ... a. cot; (p0)] (1-16) r2 vrad, 4TTa/a pz for pdS 2 a- -<< 1 , p1, 2- -q (1-17) jp2 k 'T- -10- With these conditions satisfied, the current, although non-uniform in 0 is confined to a thin layer near the surface much as in the case of an isolated cylinder. The power dissipated per unit length in the mth conductor is then a on= 1 f Jmz(r, 0,z)I2rdrdO 12 z) q L ( [1 4 1 EI a i Watts/meter (1-18) p-- I If the cylinders are now made perfectly conducting, the current on the mth cylinder will be of the form. I (z) I (z) q Kz(0, z) ) gM(t) + a Cos (PO) m fa MZia E mpp=(1-19) where g n(0) is the normalized surface current density. Using the approximation expressed in equation (1-4), the power loss per unit length for a good conductor expressed in terms of the coefficients a for themp perfect conductor is p qs1 (z)2 I (z)R 1 2 S 1 )(s)adO .-- [I 1 ±am 0 r p:.1 Watts/meter (1 -20) For large values of a/d5 this expression is a good approximation to the correct relation equation (1-18), that is LI a ~atSMP a/d >1-21) P -1 , ~provided pds 2 << p =, 2--q (1-22) 2 s The first term in equation (1-20) is the power loss in the rth conductor due to the net current I in that wire. This is the normalm skin effect loss. The sum in (1-20) represents the loss due to nonuniform currents induced by other wires in the system. It is the additional loss in the mth wire due to the proximity effect. Since the coefficients a in themp sum are a function of the net currents I1 in all wires of the system, the 1 2equation for P cannot be written as P R ( I ) if R is to be onlym m i" m m a function of the physical parameters of the system. As a result, the usual circuit definition of the ohmic resistance of each wire (Rm = m/-Im) makes no sense. When all conductors carry the. same total current at each cross section the ohmic resistance per unit length of the system of wires is a useful quantity. Using the series definition of the current (I-19) the ohmic resistance per unit length of a system or n parallel wires is given by -J2- R E R n [1 .l ' amp2) Ohms/meter (1-23) m= p= I If the separation between conductors is large enough that each cars be considered as isolated from the others, (1-23) becomes .Rs9 y-_ = n Ra Ohms/meter (1-24) The additional ohmic resistance per unit length due to the proximity effect is then Rp = R- R° = 4a am2 Ohms/meter (1-25) m=l p=1 Normalized quantities are useful when comparing different configuraticns of conductors. A q - R1 + lal2 (1-26)R. 0 R 0 m (1-27) rn-1p= 1 In the present analysis a smooth conductor with a uniform surface resistance is assumed. Recent research by A. Sanderson [18] indicates that surface roughness in the form of scratches transverse to the direction of the current can significantly alter the equivalent surface -13- resistance. The type of wire used in practical applications is usually formed by a drawing process, such as drawn copper wire. Surface scratches are in the same direction as the current flow and are expected to increase the ohmic loss much less than equivalent transverse imperfections would. Calculations using Sanderson's theory indicates that surface roughness can be ignored at the frequencies of interest (< 100 MHz. ) in this study. 3. Formulation of the Integral Equations for the Tr&sver-se Current Distributions Consider each of the long, parallel cylinders id Fig- 1-1 as being perfectly conducting. The surface current density on the Ith conductor is then K (O', z1) g 1 g1() f(zl) I = 1, z --n (1-28) The dimensionless quantity g1(O) is the normalized surface current density. In (1-28) the same z' dependence f(z') is assumed for the current distributions on all cylinders. The conductors are composed of three sections; the length z-d<z',<z+d and the two end sections z+d 4z' h, -h 4z' -z-d. In addition to the inequalities presented in (1-7) and (1-8) the following const-aints are placed on the length d Pod << 1 (1 -30a) 2 22 d2 >> n c (1-30b) This makes the current distribuions at every cross section along the -14- length z-d, ,z' %Vz+d approximately the same. K1(G~~,z' g 2gt(O) f(z), z dz'~zd(-1 The Helmholtz integral for the vector potential component A z(r.,z) at a point just off the surface of the mt conductor is a~7 1& 0 1 +a 9 ) ~ A mZ(r.O Z) 0- ________R dO'dz' z'=z..d i=- z-d h iOm + 91(01~) ffz') e ~ d where~ zhz+ I ~~'~ iOm]c~z r, 2 21/2 2 2 2 Lm (zz)' + r n. [(z-z') + 4(m-1)c+ r 2 +a 2 -Zar cos (0-0' + 4(m-I)c(r cos 0 -a cos 0)1l/2 (1-33) If terms of order Pdor less are neglected in the first integral and setting R I (Z-z') 2+ 4(m-1) 2C' 1/ (1- 34) in the last two integrals, equation (1-33) reduces to -15- f(Z)E R dO'dz' z'=z-d 9'=- - z- ip[(ZZl)2 +(-)2 c2 1/21 + 2v~Zf(z') E e [(z-.z') 2+ 4(m-1) 2 2]1/ dz (1-35) The z' integration in the first integre can be evaluated directly [19, p. 50, 200. 01]. The result is ?,+dJ f dz' = 2 sinh'l(.) -- 2[In(rt) ' In(2d)] (1-36) z'=z-d where terms of order n2c2/d2 are dropped in the last expression. With (1-36), equation (1-35) becomes Amzlr, 0, z) -2i(z) I01) fnlrrn)]dO' + 4fi nin(2d) + A'r(z) (1-37) The term Az(z) represents the last two integrals in (1-35). The normalized surface current density is given by the boundary condition -2ffa DA Me ~(r, 0, z) g(O) - - (1-38) •0oIf(z) 8r r= a -16- The next step is to substitute the approximate expression for the vector potential (1-37) into (1-38) to obtain I' " m 't 1 ('r-cos(0- gm (0) ' lir _ _" _ _" _, dO' r -,. + 1 -2,rcos(0-0') + n' +!? [2(m-1) (c/a) cos 0-cos (0-0)g1(0 (139 f i-. (r~n) f/rn where = r/a (1-40) and r -[4(m-1)2(c/a)2 + 2 -2cos(O-0,) + 4(m-1)(c/a)(cosO-cos0')]1/' (1-41) The first integral, which represents the self term, is indefinite when = 1; therefore, the order of the limiting and integration processes are not interchangeable. For values of r near unity the integrand has the behavior Integrand Z- 2 + (1-42) A +0 where A < '-1 << 1 (1-43) -17- Comparing this with the following definition of the Dirac delta function 0(0) -- rn (1-44) A-0 + it is evident that in the limit the integrand becomes Integrand = (0) + 1 (1-45) Substituting (1-45) into (1-39) and rearranging yields gm(0) = 1 + f (0 0 ,6') g(0 ')d0' (1-46) 0 -r 1=1 where Km (00') + Z(m-1) c/a coa) -cos (,-0') (1-47) Symmetry about the center of the system of wires requires g n+lm() gm (r-) which reduces the number of terms in (1-47) to n/2 for n even or (n+l)/Z for n odd. n even IT, g (0) = K1mn-O') g (0t)dO' + 1 + 1 m VT M,/I - m IT [Of 7rI= 1 mff )- S•[KmPI(OfOf) + K m, n+l.-I10,7r-0') ] g,(0')dO' P m = It 2 -- n/2 (1-48) -18- n odd hmmgi(0) = him} K n+.(0,f-01) gm(0I)dO' I+ m'-l M.m n' = - [K11.1 O'l Kn.lom- O']fO'd' n , (n I)/2 I0'O)gln+lliO)dO (1-49) where h(m) 1 m j (n+l)/2 = 0 m= (n+l)/2 (1-50) Equations (1-48) and (1-49) represent, respectively, a system of n/2 and (n+l)/2 coupled integral equations whose solutions are the desired surface current densities, gi(0). 4. Solution for Two Conductors The simplest geometry for examining the proximity effect is two parallel circular conductors carrying equal currents. Exact expressions for the current distribution and ohn-ic resistance for this simple case are given in Technical Report No. 612 [16]. The normalized current distribution on the two wire system is graphed in Fig. 1-3 for vario is conductor spacings c/a. The distributions will prove useful in developing approximate solutions for systems with two or more conductors. 1.5 Co2 g(9) C/ z5.0 -vL 20 7r/4 7/72 37r/4 7 9 NFIG. 1-3 THE NORMALIZED SURFACE CURRENT DISTRIBUTION ON TWO WIRES VARIOUS WIRE SPACINGS c/d -20- 5. Approximate Solution of the Integral Equations for Two or More Conductors A. The Method of Undetermined Coefficients 1. Reduction to a Set of Algebraic Equations For systems with more than two circular conductors approximate methods must be used to obtain the current distribution and resulting ohmic resistance per unit length of the system. In this section one such method, undetermined coefficients, is applied to the previously derived system of integral equations (1-48), (1-49). As the previous analysis suggests, a trigonometric series is the natural choice for an expansion to represent the normalized surface current density. q gin(0) 1 + E amp cos (pt) (1-51) p-1 Further evidence for this selection is found by examining the exact solution for the two wire case. A Fourier analysis of the current distribution (Fig. 1-3), for the limiting case c/a -: 1, indicates that the first two cosine terms in the series are adequate to predict the correct value of the additional loss due to proximity R p/R to within 1%. For large spacings, c/a >> 1, the current distribution is of the form 1 + a1cos 0 as is evident from Fig. 1-3. This last statement is also true for systems with more than two wires and is easily understood if the magnetic field due to external currents is considered a constant over the cross section of each conductor. The magnetic field B normal to the my axis of the mth conductor in a system of n conductors would be Bmy =i (n- (1-52) 1= 1 I/rn and the resulting current distribution becomes n• a V " I n(0) 1 1 + (2 )Cos 0 (1-53) 1=1 I/m Substituting (1-51) into the integral equations (1-48), (1-49), one obtains n even ap [-Cos (pO) + K m m(Op 0, cos (pO, dO' MP m n+l -p= 10 + p [Km (OO') + (-)PK IT ] KM nos (p osm)=d)O TJ'j",n+.. (OO') + LK,(OO') +Km n+l(O0'01)] dO' I/M (1-54) m 1, 2, --- n/2 4 -22- n odd q~ amp [-c osOS P9 r/(n+1)/Z I., k- 1L h(m) J M, n-I-rn 0 I~ o~ O'd I ( / 2pO m=(n+l)/2 + ±al K , (9, O')+(-1)PK, (Blll, B')]coslpO')dB'/-: pl 7 + anl h_ Km,(n+l)/2(O,,)osZp,)j,=. h(m)Km, n+l1m(0,0,77 1O)cos)P ( 0, 0')dB' ~ O'dO P.=1 Z 9 Z'f f (n-l)/z ' + [Kin (iOO,) + K, n (09') do m -- I1,2,- --(n+ 1)/Z S(1-55) where the same number of harmonic terms q is used to represent the surface current on all conductors. Due to symmetry about 0 -?T/2, only even harmonics appear on the center conductor of a system with an odd number of conductors. The definite integrals in equation (1-54) and (1-55) are of the form iT I(L),m-fp) 1 +(f[+2(m-)(c/a)cos 0 -(cos 0 cos 0'.. 'T [4(m-I)Z(c/a)2 + 2 + 4(m-)(c/a)cos 00 ' -r f sin I si i 0') cos (O') dO (4(ni-f)(c/a) + Zcos(')) co: U,'- 2 si'n0 sin-5o -23- Appendix A contains a detailed evaluation of this integral, the results of which are 1 2 2 p [As2+ Bs + C] p= 1,2,--q I(Om-Ip) = ((1-57a) -1 2[Bs+C] , p=Os(l -s2) where s = (4(m-1)2 (c/a)2 + 1 + 4(m-1) (c/a) cos 0)1/2 (1-57b) A = cos (0 -(p -1) ) (I-57c) B = 2(1 + 2(m-I) (c/a) cos B) cos (p4,) (1-57d) C = cos (0 +(p+ I) l) (1-57e) I#.tan-I1 sin 0 )'(m-1) = 1-- 'Z(m-1)(c/a) + cos0 I I -sin 0 2(n-) (c/a) + cos 0) (rn-I) = -1,-2,-- The principle value of tan' is used in (1-57f). For the case n even the system of equations (1-54) with (1-57) becomes q n/2 q I ap [- Cos (pO) + (-l.)P1 (0, 2m-n-1,p)] + I / ap p=1 = I p=1 I/rn [1(0, rn-, p) + (- 1)1 1(0,m+I-n-1,p)] = -f I(9,2rm-n-1,O) n/2 + [1(O,rn-1,O) + I(0,m+I-n-1,o)]} m= 1, 2, --,I/2 (1-58) 24 This is a set of n/2 equations, one for each conductor, involving qn/2 unknowns (a mp) the coefficients of which are functions of the variable 0. In order to solve the system a set of qn/2 conditions is necessary. Two procedures which yield such conditions were used, the methods of collocation and least squares [251. 2. Solution by the Methods of Collocation and Least Squares In the method of collocation, the a are chosen so that equation mp (1-77) is satisfied exactly at q points 0 mk (0 <0mk 4 1, k 1 1, 2, --q) on each conductor, more specifically for n even q I amp [- csp k)+ (-I)p 2 k m- n-1, p)] cos(Pemk)+ IOk p=1 n q + I a [nk1()n"-'P) + 1)P COmk, m+1-n-1.p) 1-I p-IIm n/2 "I(Omk, 2mn'l,O) + I I(Omk, n1",O ) +I(Omakp m+1n-'1,O)] } m 1,2, --n/2 k -2, --q (1-59) With the definition of the new variables tm tin1 and s equationkp' kp' ink' (1-59) becomes q n/2 q pa 1- p: I/M In matrix form, the system of algebraic equations (1-59) is now T 11 T 12 T- T1 n/ A1 1 IT21 T22 --- T 2 n2 A2 S2 T T T /2n2- ~/- L/- (1-61)n 2 n/2 A! where a.1 -- 1 5ii (1-62c) a. i ti- t (1 -62a) of te adTmte ar rel thrfoe euton (- 1) ca e(-6b write as two seart equaion [T (a r]lS (1 -63a) A,.S -26- [T] [A]1 0 (1 -63b) where A r contains only the real part of the coefficients Re(a )and A1 nip contains only the imaginary part, In-(a ).For a unique solution ofmip (1-63a), the T matrix must be nonsingular. A nonsingular T matrix indicates a trivial solution for the homogeneous equation (1-63b), i. e. all Im(amnp) 0. The normalized current distributions gm()are therefore real quantities. The method of least squares differs from collocation in that the a are chosen in such a manner that equation (1-58) is satisfied in amip least squares sense over the interv'al 0 41- c 7r rather than satisfied exactly at specific points, namely I1 [-ipcos (pO) + (- 1)P 1 (0, 2m- n- 1, p)]j + I/ am a Ip [1(0, ni-1, p) 6. p= I 1l p4 n/2 Z + (-l)P 1(9, m+f-n-1, p)] + 1(0, Zrn-n-1, 0) + [1(0,mi-1, 0) +I1W,nm+I-n-l, 0)ljO I/rn minimum mn = 1,2 --- n/2 (1-64) Differentiating the left hand side with respect to each coefficient a p and setting the results equal to zero yields fI-cos(kO)+(-l) k W(,Zm-n-l,k)j 1 -o~O -)If,2in1 ) 0- 0 I n/2 q 1 a1I(m-P)+-)~ (,+--p) + I(O,Zni-n-1,0) II -27- n/2 + [I(0,m-1,O) +I(0, m+l-n-1,O)] dO = 0 (1-65) M/m m= 1 2,---n/q k = 1, 2,---q After rearranging terms and performing integrations, (1-65) becomes 8(k, P) -f [(- 1)P cos (kO)I1(0, 2m-n- 1,p) + (_1)k cos (PO)(0 2m..n.1, k) p=1 0=0 n/2 -(-IIP+kjlO, 2m-n-l,p)Il0,2am-n-l, k)]dO + I 'v atp I/ f [cos(kO) -(-l)kI(o, 2m-n-1, k)][I(O, m-l, p) + -)IOm+i-n-1, p)]dO 0=0 Sn/2 f [cos(kO)--_1)k 1(0, 2m-n-l,k)] I(0,2m-n-lO)r+ [ lrn m-IO) 0=0 I I/m + I(Om+l-n-1,0)] dO (1-66) m: 2f,---n12 k = p ,2, --- q which can be written as q n/2 q a ta m trr+ a, t k 5 (1-67) p= 1 1=1 p= 1 I/rn -28- The variables t, and smk enter the matrix equation (1-62) in the same manner as in the method of collocation. B. Numerical Results 1. Comparison of the Two Methods of Solution For the collocation solution, the same matching points (8ik = 0 0 < 0k 4 7r) were used on all cylinders except the center cylinder in a system with n odd. The current on the center cylinder has symmetry about 0 = i/2, ir; therefore only points in the first quadrant are needed. These were chosen to be 0k/2. Several different combinations of matching points were used in (1-59) and the resulting matrix equation (1-61) was solved for the coefficients a using a rmp standard Gaussian elimination algorithm [261. The additional ohmic resistance due to the proximity effect R p/R0 was calculated from (1-27) for various numbers of harmonic terms q. No particular distribution of points gave a best rate of convergence of R /R for all numbers of p 0 conductors and spacings. The final set of matching points settled on is k~ q even k( V 4) 2k q odd(k+ 1) k > qI (1-68) 2 For an even number of harmonics, the points are equally spaced and internal to the region 0 4 0 k 4 7r. With an odd number of harmonics, a slightly better rate of convergence was found when the set of matching points did not include 0k .7/2. -29- The least squares procedure requires the evaluation ot the de- finite integrals in equation (1-66). Due to the complexity of the 1(0, m-I, p) functions in the integrand, an exac't evaluation wa! unob- tainable and approximate numerical integration necessary. A typical integration from (1-66) was performed using three different numerical integration routines: Romberg, Simpson's rule, and Gauss quadrature. The six-point Gauss quadrature formula [27] required the least time for the desired accuracy. The interval 0 -0 < ?r was divided into k+l or p+l panels, whichever was larger, and the six-point formula applied to each panel. With the integrals evaluated, the resulting matrix equation (1-61) was solved using the same algorithm as for the collocation solution. A comparison of the two methods is presented in Fig. 1-4, where R p/R and the computation time for the 1. B. M. 360/65 computer are graphed as a function of the number of harmonic terms used in the solution. The results are for 4 cylinders with a spacing c/a = 1. 10. Least squares is the more elegant of the two procedures, converging to the limiting value of Rp/R 0 when the number of harmonic terms is less than half that required in the collocation solution. In terms of computation time the collocation method is much faster--roughly 6q times faster for a given number of harmonic terms. Thus the limiting value of R p/RA is obtained in about one-tenth the computation time needed for the least squares solution. Similar time savings are found for other numbers of cylinders and spacings. For this reason, the majority of t:he calculations for this work were do,,e by the method of LO- Culoce1tion 0.6 . Lowt Sqwon S O; 0.4 --401' C 0.2 U' Collocation -l Prcso A..1kmati. o *2 4 6 8 10 12 14 15 18 20 Number of Har~monic Termsq FIG. 1-4 COMPARISON OF THE LEAST SQUARES AND COLLOCATION SOLUTIONS FOR FOUR WIRES WITH SPACING c/da ..10. -31- collocation. Listings of the computer programs for both methods are in Appendix B. 2. Transverse Current Distributicns The number of harmonic terms used for the current distribution on a given system of conductors was determined by observing R p/Ro. If increasing the number by two produced less than a 0. 10% change in Rp/Ro, the number of terms was deemed sufficient. The normalized surface current densities g(0) for systems with 3, 4, 5 and 6 conductors and various spacings c/a are plotted in Figs. 1-5 through 1-8. The distributions for 2 conductors are not plotted, since they are identical to those in Fig. 1-3. In systems with three or more closely spaced cylinders there are both positive and negative currents on the surface of the outer conductors. These currents in opposite directions add nothing to the net current in the wire; they just increase the ohmic loss. When the spacing between cylinders is very close, currents on adjacent surface.s of two conductors tend toward equal values with opposite sign; for example: for 4 wires, spacing c/a = 1. 1, g(7) I -2 on cylinder 1, while on cylinder 2, g(Ir) +2. 3. The Additional Ohmic Resistance Per Unit Length Due to the Proximity Effect Computed values of the additional ohmic resistance per unit length due to the proximity effect R p/R for systems with various spacings c/a and up to eight conductors are presented in Fig. 1-9 and Table I-1. Calculations of R p/R were not made for extremely close spacings, i. e. Wie C/o -1.1 ire 22.079. 2.0 5. 1.0s 1.1Wie 00 0 * '92 7 -1.0 6- 2-2.0 FIG. 1-5 NORMALIZED SURFACE CURRENT DISTRIBUTIONFO FOUR WHRES IE i Wife 2 .olWire I 2.0-Z 2.0. 1.0-2 g(e) g( 8) C/o, ... a e Wire 3 c/d ." gl) 5.0 0 T/ I t 8 NI FIG.1-7 NORMALIZED SURFACE CURRENT DISTRIBUTION FOR FIVE WIRES g(e). ~o 22D 1-0- 1.0Wie 5.0 L FIG 2-8 NOMAIEDSRFCECRRNDSTIUTO FOR SIWRE -35- Table 1-1. Normalized Addi-donal Ohmic Resistance Per Unit Length Due to the Proximity Effect R p/Ro. Number of Conductors Spacing c/a 2 3 4 5 6 7 8 1.00 0.333 1.05 0.316 0.748 1.231 1.10 0.299 0.643 0.996 1.347 1.689 2.020 2.340 1.15 0.284 0.580 0.868 1.142 1.400 1.693 1.872 1.20 0.268 0.531 0.777 1.002 1.210 1.401 1.577 1.25 0.254 0.491 0.704 0.896 1.068 1.224 1.365 1.30 0.240 0.455 0.644 0.809 0.956 1.086 1.203 1.40 0.214 0.395 0.546 0.674 0.784 0.820 0.965 1.50 0.191 0.346 0.470 0.572 0.658 0.732 0.796 1.60 0.173 0.305 0.408 0.492 0.561 0.620 0.670 1.70 0.155 0.270 0.353 0.428 0.485 0.532 0.573 1.80 0.141 0.241 0.316 0.375 0.423 0.462 0.495 1.90 0.128 0.216 0.281 0.332 0.372 0.405 0.433 2.00 0.116 0.195 0.252 0.295 0.330 0.358 0.392 2.20 0.098 0.161 0.205 0.239 0.265 0.2S6 0.304 2.40 0.032 0.135 0.170 0.197 0.217 0.234 0.247 2.50 0.077 0.124 0.156 0.180 0.198 0.213 0.225 2.60 0.071 0.114 0.144 0.165 0.182 0.195 0.206 2.80 0.061 0.098 0.123 0.141 0.154 0.165 0.174 3.00 0.054 0.085 0.106 0.121 0.133 0.142 0.150 3.50 0.040 0.062 0.077 0.087 0.095 0. 101 0.106 4.00 0.031 0.048 0.058 0.066 0.072 0.076 0.080 0 3- WS 0 ~0 E U Ud a 0.4Z 2w (0 Eq. Ohi0 La-0 -37- 3 and 4 wires, c/a less than 1. 05; 5 or more wires, c/a less than 1. 10. The reason for this will be evident after a closer examination of the approximation already made. In the limit as c/a approaches 1. 0, the surface current devilops large spikes at adjacent points on successive cylinders. This is illustrated for 3 wires with spacings c/a = 1. 10, 1. 05, and 1. 01 in Fig. 1-10. For wires with finite conductivity, a change in the form of the current distribution in the radial direction is expected to accompany these areas of high current density. As a result, the radial decay rate will differ from the high frequency skin depth ds in these regions. This is basically the same idea expressed in equation (1-22). Spikes in the surface current require high harmonic content (p large) which, from (1-22), require very small skin depths (high conductivity) for the high frequency skin effect approximation to be valid. In Fig. 1-11 the resistance R/R is plotted against the number of harmonic terms used in the series representing the current. The coefficients a obtained by either of the approximate methods, unlikeMP the Fourier coefficients, are a function of the number of terms q used in the series. They approach the exact coefficients in the limit as q becomes large or, in terms of the resistance, as R p/R converges to the limiting value. For this reason the coefficients used in constructing Fig. 1-11 are those found for the limiting value of Rp /R .From Fig. I-Ii, 6 harmonic terms are sufficient to give R p/R to within 1% of the limiting value for the minimum conductor spacings presented in Table 1-1. Using equation (1-22) with 6 harmonic terms, the high frequency skin effect approximation will be valid provided Wkse 2 %stolWiro I 1.0. LOO toL o- 7,1.102.0 - do 1 --- - do p -p - ,1.10_ _ _ _ _ -.U- -" -.a -o.4" .. Re r~~-p5LI '1-.6, 1.10_ _ _ _ _ _ 0p -p 0- 1 .0 o : .0 a w d o -L i m it in g N-3,/o a 1.05 Vole 0 I I! I _ I 1 2 3 4 5 6 Number of Harmonic Terms FIG. 1-il THE RESISTANCE Rp/R0 AS A FUNCTION OF THE NUMBER OF HARMONIC TERMS USED TO REPRESENT THE CURRENT DISTRIBUTION ON EACH WIRE. THE SPACINGS C/o ARE THE MINIMUM VALUES PRESENTED IN TABLE 1-I.A. -40- a/d >> 1 (1-69a) ad1 (<111-69b) (1-9-) These conditions are satisfied by most wire sizes used in practical antennas operating at frequencies above I M z.; for example: 1/8 inch radius copper wire has the following values (3ds/a)Z Frequency (MHZ.) a/d s (l9d 1/a) 1 34 1. 1 x 10 2 10 107 8. 5 x 10-4 100 340 8. 0 x 105 The failure of the skin effect approximation for extremely close conductor spacings places no serious restriction on the usefulness of the solution, since in practical applications the minimum spacing, deter- mined by the thickness of the wire insulation, is usually within the range of values covered in Fig. 1-9 and Table 1-1. 4. Comparison with the Work of Butterworth As previously mentioned, Butterworth has calculated the ohmic resistance of systems of paralle] wires. In this section two of his formulas, rewritten in the form Rp/Ro, are compared with the present -41- calculations of additional ohmic resistance due to the proximity effect (Fig. 1-9). The first graph in Fig. 1-12 is a comparison with Butter- worth's "semi-empirical formula" which, for high frequencies, can be written as [10, p. 709, equation 53] R 1 . u a/c)2 R wn(a/c)4 + n (1-70) 2(l- 1 v(a/c()2 4 n where un, Vn, and wn are constants which depend on the number of conductors in the system. This formula gives results which are in fair agreement with the present calculations. In the second graph of Fig. 1-12, the present theory is compared with another of Butterworth's formulas, one which is often found in handbooks on coil design [11], [13], and [14]. This formula is derived by making assumptions siniilar to those already discussed. Consider each conductor to be in a uniform magnetic field due to the other conductors. With (1-53) and (1-4), the power loss per unit length in the th m conductor is .12 Rs I1 + -!(a/c) 2 ( Watts/meter Pm =i a L 2ac (=1) (-1 1=1 (1-71) 1/in and the resulting ohmic resistance due to the proximity effect becomes R (a/c)[ 1 1 ni= 1=1m 1/rn N*6 ---- Butterwothis %ei-Empirical L5 N uNumber of Wires Q5' .2 ' R0 1 N86 --mterworlhis Formula for Wires with Moderat Spacing. As Presenta8 in Terman-NRdio 1.50 Eninas' Hadoa Rp, N- Number of Wires 1.0- 0~5 1.0 1.5 2.0 3.0 Spacing c/a FIG-1-12 THE ADDITIONAL OHMIC RESISTANCE DUE TO THE PROXIMITY EFFECT- COMPARISON WITH BUTTER WORTH'S SOLUTIONS. -43- As seen in Fig. 1-12, this formula gives results which are obviously in error for spacings in the range 1 i< c/a c 2. It is applicable only in a region (c/a >> 1) where the proximity effect is of little interest, since values of R p/R are small and fairly independent of the number of wires. 5. Optimum Conductor Spacing when the Cross Sectional Dimensions are Restricted In certain applications a given number n of parallel, in-line conductors must fit within a specified length .; see Fig. 1-13. It is of interest to ask for which wire radius a, or spacing c/a, is the resistance of the wires a minimum. If there were no proximity effect, making the radius of the wire as large as possible (a = 2/2 n) would minimize the skin effect resistance. With the proximity effect present, increasing the wire radius increases the loss due to proximity and a minimum resistance point is reached where the decrease in skin effect loss is just balanced by an increase in proximity loss. In Fig. 1-13, the dimensionless quantity 27r 2 R/nRs, which is proportional to the ohmic resistance per unit length of the system of conductors, is plotted against the normalized wire radius a/I. The points of minimum resistance are clearly exhibited in Fig. 1-13 and the corresponding conductor spacings are listed in Table 1-2. Number of a/1 c/a 2n.R Conductors n R s 2 0.250 1.00 5. 33 3 0.148 1.19 10.41 4 0.098 1.37 16.07 5 0.071 1.50 22.01 6 0.056 1.59 28.10 7 0.046 1.66 34.30 8 0. 039 1.71 40.57 Table 1-2. Conductor Spacings for Minimum Resistance. 2TtR NeS 7 40- N Wires 6 a 308 5 204j Minimum Resistance 3 01s i .i .m fill. s l la aI0 0.05 0.10 0.15 O20 C.25 FIG. 1-13 THE OHMIC RESISTANCE AS A FUNCTION OF THE WIRE RADIUS a WITH THE DEPTH OF WINDING t FIXED -45- 6. Conclusion Systems of equally spaced, in-line conductors carrying equal currents in the same direction have been studied. A set of integral equations was formulated to determine the transverse distribution of axial current at high frequencies when the current is confined to a thin skin near the conductor surface. Using the integral equations, an approximate solution for the current in the form of a trigonometric series was obtained. For two wires, the approximate solution for the current showed good agreement with an exact expression obtained by a conformal mapping procedure. With the current distribution determined, the high frequency resistance per unit length cf the system was calculated for various numbers of conductors and spacings. The results of these calculations may be summarized qualitatively as follows: i. For small numbers of conductors, the additional ohmic resistance due to the proximity effect Rp/R increases either with an increase in the number of conductors or with a decrease in the conductor spacing. This was checked for systems with up to eight wires and spacings as close as c/a = 1. 1. ii. For closely spaced conductors the additional ohmic resistance due to the proximity effect can be greater than the resistance of the isolated wires. iii. When the cross sectional length I = 2 a + (n-1) c of the group of conductors is restricted, there is a definite wire radius that will give a minimum resistance per unit length for the system. -46- Oniy cylinders carrying equal currents in the same direction were considered in this chapter. With a timple scaling of the harmonic terms on each conductor tae present theory and associated computer codes could handle systems of wires with different net currents in each wire. Such a solution would be useful for making computations for multiwire transmission lines where the wires carry currents with equal magnitude but in opposite directions. SECTION H THE ELECTRICALLY SMALL MULTTURN LOOP ANTENNA L Introduction The single turn loop has been the subject of much investigation and from the practical standpoint adequate design data are avail- able for this structure [29], [29 The multiturn loop, with no restrictions on electrical size, has received much less attention. The solutions available are for the "one dimensional" current distribution and therefore, strictly speaking, only valid for loops with spacings between turns large compared to the wire diameter [30J. [311 In practical applications, the electrically small loop ic often used because it has a desirable field pattern as compared to larger loops whose patterns have many lobes. The ohmic resistance of small loops is in general much larger than the radiation resistance, thus radiation efficiencies are very low and greatly dependent on the ohmic resistance. In an effort to in - crease the radiation efficiency multiturn structures are often used. The radiating properties (radiation resistance and field pattern) of electrically small single or multiturn loops are easily derived, either directly from the integral form of Maxwell's equations [28], [32] or aw a limiting case of one on the more general analyses mentioned above. These methods are usually concerned with perfectly conducting wires and thus provide no information about ohmic loss of the antenna. -48- The ohmic resistance of a small loop is usually taken to be the same as that of an equivalent length of atraigbt conductor. This assumption, although adequate for the single turn loop, is not for the multiturn case. In a multiturn loop, the distribution of current over the conductor cross section is determined by the same effect3 discussed in Chapter I --proximity and skin effects. The increase in ohmic resistance due to the proximity effect, which is normally unimportant in large antennas, has a dramatic effect on calculations of the pvwer radiated by electrically small transmitting loops. 2. Review of Small Loop Theory The properties of electrically small loop antennas covered in the literature are briefly discussed below. For a more detailed discussion, see King [321 or King and Harrison [28]. The model chosen to represent the multiturn loop antenna is illustrated in Fig. 2-I. All turns of the loop are circular and lie in parallel planes. The straight segments of wire interconnecting the turns and the feed wires of the delta-function generator are short, parallel and closely spaced. These are assumed to have negligible ohmic resistance compared to that of the overall circuit, and to contribute negligibly to the radiation resistance since parallel segments carry equal and opposite ly directed currents. The dimension Zc is exaggerated in Fig. 2-1. The nmltiturn loop with n tarns will have essentially the sanji total current (I) at any conductor cross section, provided I:, II ii m5. M wa- 0 L CC a LCD -50- the total length of the loop is much less than the free space wave- length at the operating frequency. More specifically, I (s) ---1 (2-l) when onb l (2-2) For this analysis, the following additional constraints are placed on the wire radius a and the turn spacing c. a b , oa.<<l (2-3) n c z<.b 2 c-a (2-4) A real power equation expressed in terms of the scalar and vector potentials 6 and A for the loop antenna is Re J = Re dv -iW JAdv -iw Ids (2-5)( Jf "dv) V V V S where J represents the free current density, tj the free surface charge density, a the conductivity of the antenna wire, and E5 the electric field of the delta-function generator. The first three integrals are over the volume occupied by the loop conductor and generator while the fourth is over the surface of this volume. An e time dependence is assumed. Fig. 2-2 shows sections of two typical loop turns and the coordinates associated with them. Making use of (2-1), and assuming the transverse current distribution to be the same at th any cross section, the current density on the m turn becomes r M L z C- 0 L0 0 C') I z0l ..52- 3 J(r, B, cp) = 3r (r, 8)c (2-6) Sm Jrnep r 26 where TT a af m (r, A) rdrde = I (2-7) A r=-JTr--O The assumption of constant current also precludes the possibility of a charge accumulation on the loop turns, Therefore, =0 (2-81 With (2-6), (2-7) and (2-8), and the definition of the delta-function generator E6 = -V (s) (2-9) (the distance s is shown in Fig. 2-1), (2-5) becomes n a IV ~ f f 32J c(r, a)rdrda rnl f- r TT a1 + Re [ -i2nbw jJ n (r, a) An), q rdrd]j (2-10) 0)=-TT r=0 A is the component of the vector potential tangent to the axis of th the conductor of the m turn. Referring to Fig. 2-2, the vector potential at point A due to the current element at point B is -53- dA~rooeie°) = dA (r, a, cp) 0 eoR m j (r', ')(b+r' cost') cos (9- e') rd dd'd (Z-11) where LI MA = 4b 2sin [( -d)/2 ] +4(m-)2 c2 +r2 +r'2 + Zrr' cos ( -p' ) (2-12) The vector potential at A due to the current in all turns is then At P r' ') eiOORm1 A=l if J CP'=-Tr 8 t=." r'=0 cos ( -e') (b + r' cosep') r'dr'de'dgdp] (2-13) Introducing the condition on the donductor length described in (2-2), the exponential in the integrand of (2-13) is expanded in a power series in $oR M Keeping the first two imaginary terms in this series yields Im e Io M1 (2-14) and the imaginary part of (2-13) becomes n r T a 3 mcp T -' jC 6p =-T e=nT r=O cos(q- O')(b + r' cos cp) r'drdeldplj (2-15) -54- For the purpose of calculating the second integral in (2-10), the radiation term, and approximate value of the vector potential A is used. Subject to the restrictions on the conductor radiusmCP expressed in (2-3), A is approximately the vector potentialMCP that would exist on the surface of the wire with the loop current I located along the axis of the conductors. King [33] discusses the validity of this type of approximation when used in calculating the vector potential. With this simplification, (2-15) reduces to the following n T, "B3R ~Ib RT3Im(A 7- - m cos(Cp- cp')dcp' (2-16) CP -TT where m = {4bsin [(0- 61)/2 ]+4(rn-A)2 c +a2 (2-17) and (2-10) becomes n TT a 2 L b1F0 0" 0 R 1 cos(, -')d(28) r';valtating the secotid integral, (2-.18) becomes IT a Vo-I2r mprdrdg + ?()-,fn2 ob~2 V0 0.f~~~~='T r =0 + O)rre I [ROhmic + Rad.(29) where the two terms on the right hand side of the equation are identified as the ohmic and radiation resistances. The radiation efficiency of the n turn loop is now 244 RRad. 20 n2 ob (2-20)E 0 24EA R + =R Rad. ROhmic 244 20Rm n 0 0b + ROhmi c This simple form is a consequence of the constant current assumption which makes the ohmic and radiation resistances appear as circuit elements in series. 3. Transverse Current Distributions To evaluate the expression (2-20) for the radiation efficiency the transverse current distribution is needed. If the skin effeci approximation applies, a/ds >> 1, the ohmic resistance term in equation (2-20) can be replaced by n T a 2 n ZrrMW (r, 0)=bR s,,,)rdrdel 2 7m=l- rom- -- TT r=0 (2 gmcp (de I'-56- -1 where g (o) is normalized surface current density oil',-ILo equivalent perfectly conducting loop. Using a procedurc similar to that in section 1-3, the transverse current distribution g (0) can be derived. The integral for the vector potential component A at a point th(r, 0, Cp) just off the surface of the m turn is o n g niep 8rr MA CP =-TT =I-TI C-S11" --- CP doZ ,.cos( p- de'dcp (2-Z2) where R2 2 2 2 2 RmL = 2b + 4(m- c + r + a + 4(m-)c (r cosO-a cosq') .4 -Zb (r sing + a sing ') -Zar cosacosq'] -2[ (b -r sing) (b -a sing') cos( '" =Jq- p cos(ep- D)I (2-23)JS compared to unity. Dropping terms of this order, (2-22) becomes VrT Loib ( gj(') cosllb- q) Am(r, 0, W1) ~- r J L- 'pd (2-24) mCP 8r f .' TT O'-TT 1A q -p c se -) This is equivalent to considering the quasistatic fields as the primiry factor in determining the transverse current distribution. The integration with respect to ed may be expi.zssed in the form -5?- Ib n A r(r, CP) KIk) Iy (k de Z ~:~:-' £ where K and E are the complete elliptic integrals of the firs and second kind [3,1 ]. The modulus k and complimentary modulus k' of the elliptic 'ntegralx Rre p +q(Z6a) / (W I -k (2-26b) // / Subject o tle restrictlon. .mposed "a/(2-.) and ;Z-4) (k')2 (4.2"F/I+o-- rO(1 .L) + ] where- r- t2 +f, 2 2 r2 -"" n- mc(r con -a co--,') Zar cos (2 -') ]= (2-28) %22Since rmL is of the order of 2(m-l)c, (k')z is a small quantity (k) = 0 <(M-)C « (2-29) and the power series representations for K and E are useful [35]. -58- K2 9 7 .14K Tk .L '- + -,-- , "--- (2-30a) F, 1+ -y k + k3 +.4~ -~ - -,.. in+(-) {2-30c0 -ubst cuting -ne above serie ,, (2-25) and dropping all terms s- All corn-ared to uniy a/b, (ri-1) c /b or less, the integral ir the vctor pot,. nti 1 become a ,ioi n (r, I [r,,,) lnl rn81BT (Im j +4vnn [ln(8b)- 2] (2-31) Ex'ept for a ierm with only z dependence and an additive constant, this expression is the same as that for Amz(r, i, z) in the equivaletit system of parallel, straight conducto-:s, eq.ition (2-30). Due to the syi:,-metry already assumed in th-s F rolblem. only fhe A component. of the vectoi potential is inw.Aved in the P.r."P.o- .... ;, .-rfac~e curvent density, g(. .2() ZA (r, q, cp)9 ,(0) M14 -2 (-Z r --a With (2-31) substituted into (2-3Z), the resulting equation for the current density is identical to that for the straight conductors (1-39). Subject to the inequalities pre3ented in equations (2-2), (Z-3) and (2.4), the transverse current distributions on the loop turns -59- and the ohmic resistance per unit length are thesame as those for a system of parallel, straight conductors which have the same wire radius and spacing. 4. Radiation Effictency With the restlts of the last section and equation (2-Z0), the radiation efficiency of an n turn electrically c-nall loop is 2 24 420v2 n 0 b= (2-33) Rearranging terms, the efficiency becomes A.4. 1 [(2-34) I + - ------- ~ rifb') a' 1 where a' and b' are the radius of the wire and the radius of the loop normalized to the free space wavelength, fMHz is the frequency in megahertz, and ar i s the ratio of the conductivity of the loop wire to that of copper (ocu = 5. 8 x 107 / ohm-m). In Fig. 2-3, the efficiency is plotted as a function of the dimensionaless quantity (b)3 a/ MHzr and the number of turns. The dashed iines are for no loss due to proximity (R /R = 0) while the solid li!.eB include the proximity effect for a spacing c/a = 1. 10. For most practical applications, these two lines will give an upper and lower bound on the efficiency obtainable with various turn spacings, Neglecting the proximity effect can lead to large errors in the calculation of radiation efficiency. For example, from Fig. RAO'ATION EFFICIENCY PERCENT 11111-11 I I1 III lii' \\ \ \ \ \ \ \\ \\% \\ \ \ .\\ \ \ \ \ \ \ N \\ \\\ ~\ \ \ =bA_ .. \\ % • - \\\\\ I"\c\\ el 14 e.Ch. II t i -61- 2-3, without the proximity effect, the calculated efficiency of a three turn loop can be larger than the actual efficiency of an eight turn loop of the same size with close conductor spacing (c/a = 1. 10). When the loop is used as a transmitting antenna the radiated power is directly proportional to the radiation efficiency. Neglecting the proximity effect can make the computed efficiency for a small loop in error by a factor of two or larger. thus errors in the calculation of radiated power can be as large as one hundred percent. In some applications a constraint is placed on the volume the loop antenna can occupy. If the depth of winding I is restricted to a value much smaller than the diameter of the loop (I << b, see Fig. 2-4) the results of section I-5 can be used to optimize the efficiency. With no proximity effect, the maximum efficiency is obtained when a = I/2n and is independent of the number of turns n. E (2-35)A 1. 70x 109 fM1,, 1+ 3 If the proximity effect is included, the antenna has optimum efficiency when the turn spacings tre those presented in Table 1-2 AE- 8l14 T Z. . (2-36) + "n(b,)3( RADIATION EFFICIENCY PERCENT 0 0 0 I I I i lI I I I I 1 1 47 o'"I M0 GD ;<v 90 0)w 3-4 4A- c I I~ I i l I I -63- where f' = */I and (2 r1R/Rs) is the value given in Table 1-2. Both equations (2-35) and (2-36) are graphed in Fig. 2-4.I I 2rrR\ As the number of turns is increased, the term- 2 Rs in (2-36) increases, causing a decrease in the efficiency. With the antenna restricted. to a volume of this shape, it is better, then, to optimize the wire size rather than to increase the- number of loop turns. At a first glance, this last statement seems contrary to the common notion that increasing the number of loop turns increases the radiation efficiency. It must be kept in mind that one usually speaks of increasing the number of turns while keeping the wire radius and spacing constant, so-ne- thing which is impossible to do when the depth of winding I is also fixed. Power is usually supplied to electrically small antennas through a suitable matching network. The components in the matching network often introduce losses as large as the ohmic loss of the antenna. The overall radiation efficiency of the antenna-matching network combination is then E = EA EM. (2-37) where EA and EM are the efficiency of the antenna and matching network individually. In this chapter, only EA is considered; for a discussion of matching network efficiency, see Wheeler [36]. -64- 5. Conclusion The analysis in this chapter has shown that the results obtained for the ohmic resistance per unit length of a system of straight wires are applicable to the electrically small multiturn loop when the depth of winding of the loop is small compared to the loop radius, (nc) < < b2. Two separate calculations of the radiation efficiency of small multiturn loops were made: the first includes the added resistance due to the close proximity of turns and the second neglects all proximity losses, i. e. considers the ohmic resistance of the loop to be the same as that for an equivalent length of etrzight conductor. A comparison of the results for these two cases indicates that the proximity effect is an important factor in making accurate calculations of radiation efficiency, especially for loops whose efficiency is below 10%. The problem of optimizing the radiation efficiency of an electrically small loop confined to a fixed volume was also examined. The special case of a circular, multiturn loop restricted to a volume whose 41epth is small compared to the loop radius ff <b) was treated. For this geometry there is an optimum wire radius which gives maximum radiation efficiency for a given number of loop turns. With the optimum wire radius used for each number of turns, the radiation efficiency v\'as found to decrease with an increase in the number of turns, indi- cating that, from the efficiency standpoint, it is better to optimize the wire radius than to increase thot nuniber of turns. -65- The change in the transverse distribution of current due to the proximity effect will also alter the loop inductance. The inductance of the loop, however;. does not have to be known to a high degree of accuracy in no'Yst applications, since it is usually made to resonate with a variable capacitance in a matching network. 41r I+ SECTION li EXPER IMENTAL INVESTIGATION L Description of Experimental Avpsratus To verify the results of Section I. experimental apparatus was constructed for measuring the transverse distribution of' current on a system of parallel round wires, see Figs. 3-1 and 3-2. The parallel wires are modelled by 34" long copper tubes inter- connected with wire braids so that they carry equal currents in the sam- direction. A 100 Watt, 100 KHz transmitter drives a current of the order of 1-2 Amps. throtgh the model, which for matching purposes is fed in series with a S Ohm load. The current distribution is measured by sampling the transverse magnetic field with a sniall loop probe moanted on one of the tubes. This tube has plugs fitted with beryllium copper finger stock at both ends; these maintain electrical contact as the tube is rotated (Fig. 3-3 ). The voltage at the terminals of the loop probe is Fmplified and metered using a General Radio model 123Z-A Tuned Amplifier and Nuik Detector. At 100 KHz the t1' copper pipes are about 200 skin depths in diameter; thus the axial currents are confined to a thin layer near the outside of the tube. The tubes are also about 20 skin depths thick, so they are electrically equivalent to solid conductors. To maximize the angular resolution oi the measured current distribution, the radial dimension of the loop probe Preceding page blank 24 -J~z 494 00 ILI Ld 0 S I We 4- - - _ 4 FA 0 9r-~*~' - O ____ I V 4~ K A i 4II 'I ~& U z 0 -*~ ft -I 0"1 ~ ' w~e1 ~ -J I- j.~I ~ 02 [11 2 IL F -71- was made as small as possible (0.0501). Sirme the fields are fairly uniform over small lengths near tLe centc:r of the tube. the axial dimension of the loop could be a few inches lnug. Using j the theorv of Whiheside sad ICing [37] the voltage at the terminals Iof the rectangular loop when the tube carries a total current of one Ampere is 4 z z[(z-xo Rl) .o, IV' ~~K = .~l-L[1L 0 volts (3-1)I! 3.3 U0 + Amp. Xo = 6. 38 I0" [,1. r +In. [ra,I + Z(r- D-L 2. zi aj - where Jr. fa are the radial and axial dimensions of the locp in inches,, r the loop wire radius in inches, and R- the load impedence at the probe terminals which is about 50 K Ohms for the G. R.jY Z-A. From ( 3-1 1, the 3" x 0. 0S0" leop probe constructed of 28 gage wire provides a 0. 6 m voltlAmp signal This is more than adequWe', for metering on the G.R. 1232-.A, since it hat a maximum. sesiftivity of 10 g Volts for a full scale deflection at WV) KHz. For rigidity a polyfoam support was placed between the loop and the tube (see Fig. 3-3 ). 7nitial measurements indicated that the metering circuit waa picking up a very strong signal induced by the large loop formed by the tab.-s and interconnecting wires. To eliminate this interference, the meter was completely enclosed in a -72- copper box and all cables used were doubly shielded. The G. R. IZ32-A Tuned Amplifier and Null Detector was calibrated at 100 KHz using a pair of Hewlett Packard precision attenuators as a standard. Fig 3-4bis a schematic of the circuit used for the calibration. The linear scale meter reading is plotted against the attenuator setting in Fig. 3-4a. The small vertical lines indicate the experimentally deterinined points; a * unit reading error is assumed. The solid line was constructed by fitting polynomials to the experimental points over three ranges. The polynomials in the form csed to correct the experimental data are v ,63V< V <130 (3-3a) z -4 2V = V + 3.5-4.8 x 10- (V -16)-5.7 x to-(V -16) 16<Vm< 63 (3-3b)in i m mn i tM = V +3.5 a 0< V < 16 (3- 3c) where V is the meter reading and V' the corrected meterM M reading. This correction is apparently only a function of the meter circuitry and not the linear amplifier section of the instrument, since the same correction applies over a 40dB. range of amplifier gain. Since the theory predicts both positive and negative currents on closely spaced tub -, a method was devised to experimentally verify a 1800 phase shift in the current density. Referring to Fig. 3-1 .a small tcrrice loaded loop is used to sample the field of the l:,rge- loop for-ned bv the tub-s and the interconnecting wires. 7:is r-fert-re signal is added to t~he signal from the current probe I I;bw Fo to Eaft elelftq IC W5 I a 3w Q It aM (a)CAUSRTIIOCRV PRECISION 50-Q LOAD (b) SCHEMATIC OF CAUBORATIOt4 CIRCUIT FIG. 3-4 CALIBRATION OF THE G.R. 1232-A TUNED AMPUFIER AND NULL OETECTOR -74- in a resistive summing network and then metered. The phase of the current on the tube is determined by noting if the signal from the current probe adds to or subtracts from the reference signal. 2. Correction for Interconnecting Wires In addition to the net currents in the tubes, three other current elements influence the current distribution on the tube cross section. They are currents in the horizontal and vertical interconnecting wires and negative line currents which represent the absence of a continuation of the axial current beyond the ends of the tube. Referring to Fig. 3-5. these currents can be treated as filamentary elements since each is at a distance from the probe which is large compared to the tube radius (a = 20a, w = 60a). As a result, their effect on the transverse distri- bution of current is additive in the sense that it may be subtracted from the measured data to obtain results for direct comparison sith the theoretical distributions. The vector potential at a point (r, 9, z) near the center of ththe m tube is iT mr,o.z ) AA + YA = A g- r $mmz MY L Ti'~ S ln(rm ] dqtz'd + A' (z) ' d-' od 0=0 d- + ._ _ dj (3-4)Z'Rm Rm Rm .f= M=1 m13 m=1 TzI R i n* RM1I M12 Rmli (MI~I FIG. 3-5 INTERCONNECTING WIRES USED IN THE EXPERIMENTAL MODEL -76- Rm I = [(w-rsiS)*2 + (Z(m- c + r cose) + (--)2] (3-5) Rm9Z= [(z-z)z + (z(m-Qc + r coss)2 + (r sing)]Z (3-6) t = (s-z) + (z(m-J$c + r cos ) + (y-r sine) (3-7) [(s)= + (Z m-J)c + r cosp + (y-r sine 2 (3-8) MA 4~ + (Zym4 sie)1 g~e(a) is the normalized surface current density induced in the cylinders by the three external current elements. It is the term which must be subtracted from the measured current for comparison with theory. The following boundary condition relates Am and gmc" m-gmI~a) -2a- 2 sine 1v r3-a I 2r 2A ~ Substituting (3-4 into (3-1% yields f Im gMC(') r" cos (a'-) del mc f -r + T co (-') 1 + i Km 1(1, 0) g~c (0) dg 7...) (ai/w + 2(m-( c /w) Erz =-0n t,--mss z =0 m1 no W n z =s A=(My=O x (i -77- where 2 2 = [(l-(a/w)sine) + (2(m-j)(clw) + (a w)cosO) + (z'Iw)2] (3-11) r' = [(z'I/s)2 + (2(m-1)(c/s) + (a 8) cosO) + (a/s) (sina)Z (3-12) m12 r' :I + (2(m-1)(c/s) + (a/s)coso)2 + (y/s -(a/s)sing) ]z (3-13) mt 3 The first integral in (3-10) was evaluated in section 1-3; the other integrals are a standard form [19, p. 50, 200. 03]. Performing these integrations, (3-10) becomes gmc(E) = M--lf' i Ki e' ') gj(C ')d ' -2(a/w)(s/w) (a/w + 2(m-L)(c/w) cose -sin ).(I (.a /w)s9in )z + (2(m- t)(c /w) + (a/wlcoso)][ (1-(a/wlsino)z + (2(m-A n .... ... 12 -Z(a/s) (a/8' + (m-)(c/s)cosC) (c/w) + (a/w)coso) 2 +(/w)2 [ (2(m-t)(c/s) + (a/s) on) + ) zZ ] siigzI +(2(.. /s) + (/) oe)Z + (a/s) , +Z(a/s)sinq . (sing) / [I + (2(m-t)(c/s) (als)cos4 (w/s)(I -(a/w) sine + (LI + (w/s) (-(a/w)sing) + (2(m-t)(c/s) + (a/s)cosn)212 (a/s)sinq z (3-14)+, 1,,(,,-,)(c/.) + (a-/. o. q- + (a/.)ina,, ) -76- When terms of order (a/s)2 or less are dropped, equation (3-14. reduces to L. K ISiy Kfwgj(O4d2- + Z(als) 1. .. (sw(si -Z(m-$1(c/w)cosA [+ (slw)z + 4(m-Zlc LZ ) '(-I+ 4(m-A" (clw) 1 + ZI as (als + A4m-t)(CI. cosO+ ) -77 + 4Nm-)al) +4(m-oi)(cis) + 4Im-.lcls)lals) + [+4(m-Z(cls)Z ( ) If the interaction between the induced ctrrrents on the tubes is ignored, that is, each tube is considered as isolated from the others, the integral in (3-151 disappears, and a first order approximation for the current results. n zMC/() Z(a/s) ~ [1+ (si1w) + 4(m-~ (c /w) 1 (s/w) 2 i -Zm-hIOC/w)c.2!i + sina f I + 4(m- 1)1(c /w) I 1+ 4(m- L) Z(CleY"[ ~~(/ + 2(/s m-ALcst'os - 2(a/9 [(a/a) z+ 4(ni- (C /) + 4(m-A(c/s)(a/s)cose f jI _ 1 f (3-16) +[l (,-L)_cis) ] The three terms in (3-16) are due to the curzents in the vertical interconnecting wires, horizontal interconnecting wires and axial tube extensions, respectively. The currents induced by the horizontal wires are the major factors since they are about 2.(wis) 1 10 times greater than those due to the vertical wires and, for large spacings (cls) = I, at. least 3 times greater than those due to the tube extensions. To solve for the current in the complete expression (3-15), wkich includes interactions between tubes, a trigonometric series is posutlated for gmC(6). Since the average value of gmc(0) is zero and'it is not symmetric about the lines O: or n/2: 304Z the series has no constant term and contains both sine and cosine terms. q gmc(() = [a cos(p4) + b sin(pO)] (3-17)mc [mcp mcp Substituting (3-17) into (3-15), the following results are obtained n even q -11 T amp [-cos(pa) + (-fr K(, , d) cos (p&)d'JE m, n+l-m q TY +__ bmP si'O K(t3, o') sin(pu')d' bmc41 m, n~l-m -go- +~ ak ii-~[K (13. + (..E)i' X ' (6J0 cos(pO'3d& n/2 q + E -jJz 9g -(-l)P K (9. ' sir(pcfldf P; bc[KM, ni1-.t nn [I 9w (al) ( [(a/u) + 4(m-4 (c/) 4r-( s + 4(m-4 )r. mfl, 2, ---n/Z n odd q a osp.J, m f(n+1)/2 + ~m P~e'p ,10 mn = (n+11/Z m, n+t-m E t2(p+l)e, mr nl / sjnjp5)d9 +I& 1, -- 81- I mivw.Al~r gin J [K (9 ' ()P K6e')~l')g + a, /IT ~Jn, m, rK(. .) .lPK(9 ' I + j a1 p =-TIl (h(Jfl)K( ) cos(Zpq,')de'J + > in, (n+!)I A cp, 6=-r ~J =I n K (a. ~ sin(2p+1) e)d9' I =-Z(als)1 m, (n+1) /2 1 [+(s + 4(in-4(i) (s/W sinO -2(m-.Q(c/w) cosni + sine 3 n + Z(a /s) (a/s + 2(m-.q ic/s) COBB) A -- [(a/s) + 4(m-1) (cls) + 4(m-jQ(c/s)(a/s) cosol 1 + 4(m-J)(2 S m =1, 2, ---(n+l) /2 39 -32.- The dcfinitc integrals which appear in (3-18) and (3-19) can be reprexemted by two forms. The integrals contai. ng cos(W ) terms are thne same as tle integrals deoted by ](!, an-I. p) in section !-5 and evaluated in Apperudi A. The integlals in-olving sin(p!') terms are of the form m- .p I I + Z1m-jO(cla)cou -(C -!0s cos. 4(m-1) z(cla) z + +4(m-Lt)(ca)cos2 -(4(m-E)(c.'a) 2 cos(') cosG' -Z sine sins' I p = lZ --- {3-2Ob An evaluation of this integral is in Appendix C, the results t.-f which are I' (,m-Lp) =(I [A's +B's +C' 13-21a) where 2 2£(4{r.-1) (c/a) + I + 4(m-l)(c /a) cosu) (3-Zlb) A' = sin(a -(p-1) ;) (3-Zc) B' = -2(1 + 2(m-l)(c/a) coso; sin(pj) (3-d21, C' = -sin(9 + (p+l) t) (3-Ze) (Z(m-L)(c!a) +cosq (- !' =sinpI ' (3-21f) 2(mI)-(c/a) + coo ) (m-L) ---2, -- The principle vlhte of tan-I is ustd in (3-21f). -S3- With 13-Z1). the sYst*= Of equati*--S (3-18) flor th~e case a wven becomes q 012 q -a f -cos(p-) (-1)p Ig Zzi-M-t 9 P)l + acp .12 q I(9, Zmi-n-1, p)] + -b fig~ ~gP) A n (-Op I fig. M+A-u-1, viJ z Z~aIz) j01 1 s 1w) [ sine -m*(c/w)Colin*~~~ I + sw 4(m-£)(l)J singz.... + 2(als) Ws (l+ 2(M-lt)(cls) Cosa) I1 4(m-O !c/l)(/sl 4(rni-) (C is) j ___________)Cosa (c/9)__ __ rn 1,)/2(-2 -84- SImilar results are obtained for the case n odd. Formula (3-ZZI represents a set of n1Z equations rclat ng the qn unknowns. This system of eqations can be solved by either of the two approxiinai mmethods disccssed in section 1-5, methods of collocation and least sqcares. Appendis D contains a listing of a compater program whicb solves for the coefficients amcp and bMC by the method of collocation. The qn natching points in the intertal 0 Zvare chosen as k(-, n k=l, 2o-q (3-23) on all cylinder& except the center cylinder in a system with n odd where the oints are ? k=!. 2--q (324) Examples of the correction currents gm awe plotted in Fig. 3-6. Both the fall correction current (3-17) and the first order correcti.on current (3-15) are shown. A comparison of the two curves indicates C -the interaction between induced currents on the tubes m. ust be included in any accurate expression fcr the correction current. 3. Results of the Measurements Current distributions were measured on systems with up to six cylinders and spacings ranging from c/a = L 10 to c/a = 2. 50. After correcting for meter calibration, the measured values were normalized. The procedure for n-.rmalizing was first to measure 0 0i 20 -0 SIX WIRES C/o1.0 FIG. -6 NOMAUZD CURENTS FULLTH CORRtECTION TR-IRS REC a o K TR_0311 * TWO WIRES C/o 1.10 FIG. 3-6 NORMAUZED CURRENTS FOR THE CORRECTION TERM -86- I the current distribution on the system of conductors with the meter gain held constant and a known current flowing through the model. i The system of conductors was then replaced by a single conductor and the current distribution measured with the meter gain and current through the model the same as in the previous measurement. The normalized currents on the multiwire system were obtained by dividing the measured currents by the average value of the measured currents on the single cylinder. The measured currents, with the correction current gmc subtracted wit after they were normalized, are compared with the theoretical distributions in Figs. 3-7 to 3-II. The circles about the measured points indicate the range of error ( + 2 scale units) associated with the repeatability of the measurements. The measurements are in good agreement with the theory. The minimon spacings used in making the measurements were restricted to c/a = L 10 for two wires and c/a = 1. 25 for three or more wires. For three or more wires, the currents at adjacent points on consecutive cylinders are quite large when the spacings are small. The radial dimension I of the loop probe is a significant fraction of the distance between cylinders; for example, when c/a = 1. 10 the gap between the cylinders is only three times as large as fr. As a result, the loop responds to the currents on both cylinders giving an erroneous interpretation of the current density. The problem is not as severe for two wires since the currents at adjacent points on the two cylinders approach zero as c/a goes to I. ,] c/bliAl I1" 1 ----Theoretical 0 Measured points I0 9 w/2 Vr FIG. 3-7 MEASURED AND THEORETICAL SURFACE CURRENT DISTRIBUTIONS FOR TWO WIRES 2.0 Wire I. c/o 1.25 ,-ire I, c/o x-2.5 glO) /Wire 2. c/b 1 .25 Wire 2, chm- 2.5 ------ Theoreical 0 Measured points " A° ) FIG. 3-8 MEASURED AND THEORETICAL SURFACE CURRENT DISTRIBUTIONS FOR THREE WIRES V g(6) Wire 2, c/a:1. 25 Wie2,c/b1. Theortica 0 Meaured oint 172 1 FIG. 3-9 MEASURED AND THEORETICAL SURFACE CURRENT DISTRIBUTIONS FOR FOUR WIRES Ir Theoretical 0 Measured points 3V 2C7 FIG. 3-10 MEARR DAN THfORSTI% ~RA Ii 0 2. ire 2 g (9) _- Q Wire 3 0 7r/2 r -Theoretical 0 Measured points 0 1 2 2c FIG. 3-11 MEASURED AND THEORETICAL SURFACE CURRENT DISTRIBUTIONS FOR SIX WIRES -92- 4. Conclusion An experimental apparatus was constructed to measure the transverse distribution of current on systems of parallel conductors. After correcting the measured data for equipment calibration and extraneous sources, good correlation between theoretical and experimental current distributions was obtained. Acknowledgement The author wishes to thank Mr. Victor Richard of the Microwave Physics Branch, Baffi.tic Research Laboratories, U. S. Army Aberdeen Proving Ground for his interest in and support of this project. A-! Appendix A Evaluation of the Integral I(0, m-1, p) I I + 2(m- 1)(c/a) cose -cosjenrp')]cosjpn')d-' " 4(m- L) (c/a) + 2 + 4(m- )(c/a)cos -(4(m-i) (A -1) . (c/a) + coso) cose' -2 sing sina' Using the trigonometric identity cos(A 4- B) = cos A cos B -sin A sin B to combine the cose' and sino' terms, the denominator oi the integrand becomes 2D r A s + +2s cos(O' +) (A-2) where S (4(m- .)2(c/a)2 + 1 + 4(m-.)(c/a) cosA)a (A-3a) t sinA n-tan 2(m-A)(c/a) +cose (m- 1) =1, 2, --(A -3b) = -sing 2(mrn )(/a) + cosO (m-L) = -1, -2, -- The principal value of tan1 is used in (A-3b). The quantities s and t are identified with the geometry of the system in Fig. A-I. Applying standard trigonometric identities the numerator of the integrand is expanded, giving N = A cos[ (p-l(4' + i)] + B cos [ i' + i)] -C cos [(p+)(a, + E] -sin (p-1)(0 , + ,i)J + F sin [p(a' + ,)] -G sin [(p+l)( ' + *)] (A-4) I, -94- where A = coo(O -(p-1) *) (A-5a) B = 2(l + ?(m-L)(c/a) coso) cos(pt) (A-5b) C = cos(g + (p+l) ) (A-5c) E = sin(g -(p-) (A-5d) F = 2(l + 2(m-A)(cla) cosa) sin(pt) (A-5e) G = sin(O + (p+l) ) (A-5i) In terms of the new variable (tft) equation (A-I) is iT I f -Acy..P-A)oo[ -)&] +B cospI1I, m-s, p = 1 ++2scos() C cos [(p -) ] +,Esin []-F sin [(p + )j]da' (A-61 The sin [( ) ] terms integrate to zero and the remaining terms are in the form of a definite integral which is readily evaluated [19, p. 219, 858. 536] cos(Ag) Hjp '1 -IF, . I +'H cs d = z 1 +HZ) , p (A-7) Substituting (A-7) in (A-6) and rearranging, I (a, m-1, p) becomes 1 2 (1-s )(,P+l [As +Bs +C], p = 1, 2 ---I(1, )(-s, p1 = -1[Bs + 2C] p 0 A ) s(1- s2) LX | t; j g 9 m ‘me 9 20(m-t) Po FIG.A-I ' j qrl 0I7 Appendix B Listings of Computer Programs This appendix contains two computer programs written in Fortran IV language for use on the I. B. M. 360/65 computer. I Both programs compute the coetficients amp of the trigonometric series for the normalized surface current densities, gin' and the normalized resistance per unit length, R/Ro. The input/ output formats for both programs are identical and specify the following parameters: the number of conductors n, the number of harmonics q, a-d the spacing c/a. .d p b Preceding page blank i '4,I C FUaKTMAN~ IV AROGI4AN FOu4 SOLUTION BY Tl-E METHOD OF COLLOCATION C C C rTS Pk0%O.RAM USES THE METHOD UF COLLCCATION To SOLVE A SYSTEM OF EQUATIONS C FO. TilE LOEFFICIENTS OF TRIGONOMETRIC SERIES. THE SERIES REPRESENT THE CNOUtMALILED SURFACE CURRENT DENSITY ON EACH WIRE IN A SYSTEM OF NW EQUALLYI C SPACED, PARALLEL* PERFECTLY CONDUCTING WIRES. USING THIS CURRENT ANCAPPROXIMATE VALUE LF THE NORPALILED 1-IGH FREQUENCY RESISTANCE U.F THE SYSTEM4CIS CALCULATED. THE NUMI3ER OF HAkMONIC TERMS USED TO DESCRIBE THE CURRENT ONC EACH wIRE IS NH. THE RATIO CA IS EQUIVALENr TO THE SPACING BETwEEN WIRECCEN~TERS DIVIDED BY THE WIRE DIAMETER. FOR NW WIRES AND AND Nh HARMUNICS THE C SILE OF TI-E MATRIX TitIj) M~UST BE AT LEAST AS LARGE AS TINWI*NH+19NWI*NH) C wHcRE NwIzNW/2 FOR NW EVEN AN) NWl=INW.1)/2 FOR NW ODD. LOGICAL LSCLVE DOUBLE PRECISIOJN Pi.THETAtaEP,CAtS0URCEtTsSWIN CCMMCN THETAtSEP#PL/.AATRIX/1c48t49) WRI TE(6ti I F(GRMATIIHI) Pi=3. 14L5S265358974300 2 READ(5931 NWNHtCA 3 FCRi4AT(IIL3XI293XO!5.3) WRITE(6#41 NiECAtNH 4 FORMAT(///35XII921I WIRESt SPACING CAzF6.3,4'9 912tIOH HARMON IICS) NW20=(2*NW+I+(-l)**(Nw+11 1/4 t4W2E=(2*Nw-1I(-I)**NW)/4 NSI IE:NW2C*NH NAUG=NH*NW2O. I 00 14 L-19NH NRI=(.W2O-1 )*NH4L C SEFTINfb COLLOCATIUh POINTS THETA=PI*IDFLOATIL)lIDFLOATN-+1)) IF(2*INH/2) .NE.tNH.Af4V.L.LE.(NI4-'i1)/2) THETA= PI*FLOAT (L) /FLOAT (NH+21 IF(2*INH/2) .NE.NH.ANU.L.GT.(Nii.1)/2) THETA.PI*FLUATILtl)/FLUAT(NH. DO 14 PsLtNA1,Nh NKWs1. tM-LI /NH IF (NW2C.EQ.NW2EI GO TO 5 IF (NKW.E4.NWZOI THETA-THETA/2.ODO 5Nsw-aI NI=NH DG L4 N=19NAUG IF (N.EC.NAUG) CO TO 11 IF (N-Nil 1,7,6 6 NLINI.NH- Nbw=NS%41 7 NSHN-NHO(NS%-i) IF I h.EQ.NSW) GO] TO 9 SEPzCA*OFLJAT(NSW-NR.E) IF (2*NSW)oEQ.4NW4IIJ) GO U bEP=CA*UFL-3AT(Nh4I-NbW-NRW) GO TO 14 Ii IM,h)=-SWINf?*t.SIH i.,U TO 14 9 IF ((.)*NSW).EC.(NW+iLJ GC TO IC SE&'uCA*0FLUA J(Nh'L1-2tRW) T (M.N)=U(,OS(OFLOAT(4SH*TiETA) )/2,ODO-(-1.0OO)**NSH*SWIN(NSH) GOTO1 GO TO 14 11 SURCE=O.OD0 12CON TIN UE[ 00 13 LO2*INW20 IF (LU?.IEQ.NRW) GO TO 13 SEP =CA*CFLUAT(LC2-NRWiI SUUKCEaSoURCE4SWIN N.SELF4i) 13 CONTINUE TI(tN)wSURCE 14 CLDNTINUE 15 IF(LSULVEINSIZE)i GO TO 24 C CALCULATING THE NORMALIZED RESISTANCE RESN=I.O NCS aNw2E*NH 00 16 NC=I*NCS RESN-URESN.I (T(NCNAUtiI)**2)/1:LCATINWI 16 CONTINLE IF (N42a.EQ.NW2E) GO TO 18 NDBaNCS-61 00 L? hNO08NSIZE RESNzeESN+I IT(NCNAUG) )**2)/FLCAT(2*NW) 17 CONTINUE I8 WRITI169191 RESt 19 FORMATI/40X23H NORMALIZED RESISTANCE 9F7.41 W141TE(6,201 20 FURNA'T(/40X3IH-THE HARMCNIC COEFFICIENTS ARE-) 00 23 Lx1.NW2C taNH* IL -14- 1 NaNH*L WRITIN6,21) L 21 FCI4MAT(51X6H WIRE PI1) 22 FGRMAT (5Xs12FIC*5) 23 CONTINUE GO TO 26 24 WttITE(6;25) 25 FORMAT(//53H THlE T MATRIX IS SINGULAR. NU UNIQUE SOLUTION EXISTS.) 26 GO TC- 2 27 STOP E NO C THIS FUNCTION SUeACUTINE EVALUATES THE ANALYTIC EXPRIESSION FOR THE OFFINITE C lilfEGIAL SWIN. DOUBLE PRECISION FUNCTION $WikN(IHAR) DOUBLE PRECISION THETAtSEPPltl#CAtPSItHBtCcEtG COMN(JN ThETAvSEP9PI 50 IF (SEP$ 51,55.52 51 PSIzOATANIDSINIIHErA)/I-2.OU0*SEP-DCOSITHETA))) 60O TC 53 52 PSIuPI-OATAN40SINITHETA)/I2.0DO*SEP*DCOS(THETA)) 53 52OUQRT4.000*(SEP)*241.COO044.OCO*SEP*DCUS(T1ETA)) A=O.50C*UCOS(THLTA-PSI*DFtUAT( IHAR-l)) Ii=(1.000+2.0DC*SEP*DCUSTtiETAI l*OCCS(PSI*DFLUAT(IHARll C=0.5OO*OCUSITHETAPI*Cfi.O)AT(I WAR.1) I It- (IHAK.EQ.O) Cu TU 54 ShI.4(A*S*2)+e*S+(.I/(I,UUOS**2*(-S)**I II4AR+Il) Idd AVS, sbIv-Ga.s~z.coscluns95 zaaS01 ING 1.15 I123 SM(x5TI1f SCLWS A SYSTEM OF N LINUSX EQUATIONS IN N UKINMiS D TWSAb GloSSIM tLIX1.7TICNV WITH CZLiP PIVOTING. OUMiE OMCISIC36 SEPoflwagS".TEPTO.ER COMM S X&ISSIU0949 41 SswlhASKgIIw i.j3 62 42 AISt. IF IIESUP.LE.LLEI G TO 42 ifIS £Ewo.49T~R GO To *0 e3 3M%.mP1 63 T1lI&MP.IJ-EPV 4400 &5 IsKPIlk 00 &5 JAKPiIOSI. 1IMSIVINAIA.EGLEA) Go TO To 00 45jz*6 W9 47 Joj*3 49 LSCLWE-.FMkSE. T- Ia 71 .1 itS-TUR.in C FORTRAN IV PROGPRAM F~OR SULUTIUN EY TIEt METHOD OF LEAST SQUARES C THIS PROGRAM USES THE METHOD OF LEAST SQUARES TO SULVE A SYSTEM4 CF EQUATIONE C FIXt THE COEFFICIENTS OF TRIGCNOMETRIC SERIES. TIE SERIES REPRESENT THE C NORMALIZED SURFACE CURRENT PEN.SITY ON EACH WIRE IN A SYSTEM OF NW EQUALLY C SPACED* PARALLEL, PERFECTLY CONDUCTING WIRES. USING THIS CURRENT AN C APPROXIMATE VALUE CF (HE NORPALIZED HIGH FREQUENCY RESISTANCE OF THE SYSTEM C IS CALCULATED, THE NUMB~ER CF HARI'ONIC TERMS USED TO DESCRIBE THE CURRENT ON C EAGH WIRE IS NH. lHe RIATIO CA IS EQUIVALENT TO TPE SPACING BETWEEN WIRE C CENTERS DIVIDED BY THE WIRE CIAMETER, FOR NW WIRES AND AND NH HARMONICS THE C SIZE OF THE MATRIX T(ItJ) MUST BE AT LEAST AS LARGE AS T(NWI*NH+1,NWI*NH) C WHERE NW1-NU/2 FOR NW EVEN AND NWI=(NW+11/Z FOR Nim ODD. LOGICAL LSOLVE EXYERNAL FltF2*F3*F4tF5 CCNON P1,CASEPSINSEPMUTSEPMIM/IqATRIX/T(28,29)//NW2ONW2ENRWJ L*NSWHPNSWH29NW WRITE(691) I FCRMATII)A PIz3* 141593 i REAO(593) Nti.NHvCA 3 FORMATIII3XIZ.3X&F5.3) WRI TE96941tW.CA*NH 4 ORMATI///35X11,21H WIRES, SPACING CAaFb.3,AH# 912,10H HARMON NICS) Z*NWI1-I)**tNW*15 1/4 NWZEsl2*NW-I41-II*#NW)/4 NSIZE-hW2G*NH NAUGNSIZE'1 00 10 NRW*ItNWZO SEPSIM3CA*FLOAT tNW+1-2*NRW) DO 10 J=LNH NRUW*(hRW-1 )*NH+J 0O 9 NSWlIvNk20 S6PMUT-CA*FLOAT INSW-NRWI SEPM4IM=CA*FLJAT tNW4+-NSW-NRWI 00 9 NSWH-19NN NSWH2=2*NSWH NCLL(NSW-1 )*Nt44NSWH DEL-O.O IF(NW2O.NE.NW2E.AND.NRW.EQ.NW2C) JI=Z*Jt I~iJ.EC.NSWHI DEL=PI/8.O IF(NRW.EQ.NW2O.AND.NW2E.NE.NW2O) 6O TO I IrifNIW.EQ*NSW) GO TO 6 IF(NSW.EO.NW20.AND.NW2EohE.NW2C) GO TO 5 T INRCW#NCOL)--ALSQI IJ*NSWHsPi) GO To 9 5 TINACGW.NCg.iL)-ALSOlIJNSWHZ.F2) 60 TO 9 I, (NAOWNCOLI=CEL-Oo5*ALSQIIJNSWHF3I GO Tb 9 7 IF(NRW.Eg.NSWI GO TO 8 T (NRCdNCOL)z-O.S*ALSQIIJlNShi,-F4) Go To 9 i8 T INRCti*NCOL)=DEL 9 CONTINUE fINRCwNAUGI=ALSUI (J. .JP5J 10 CUJ4TiNUE IF(LStJLVE(NSIZE)) (oi TO 19 vvk TEl 6,11) I1I FO~iqAT(/40x3lh-THE HAI4MCNIC LLEFFICIENTS ARE-) J)O 14 L:1.Nw20 M=NH*I L-1 )+l 12 FR4AT(3IXbH WIRE oil) IS FLR'IAT15X,12FI0.S) 14 CGNrINUE C CALCULATINGi THE hOkMALILt~l RESISTANCE RLSNI.C NCS=NW2E*NH DO 15 NC=1,NCS RESiz14ESN&( (T(NCNAU.,)*$*2)/I-LCAT(NW) 15 CUNTINUE i IF(.,v%2E.EG.Nw2O) GO. TO] 17 NC~s NC S.1 00 16 NO=NCSoNSIZE RESN=RESNI( ITINDNAUGI )**2IiFLCAT(2*NW) 16 CONTINUE 17 wsRITE(6*181 RESk j 18 FCRMATf/40X23H NORA4ALILEL; RESSTANCE tF7.4) GO TC 21 19 WRITE(6920) 20 FORMAT I//53H THE T MATRIX IS SINGULAK. NJ UNIQUE SOLUTION EXISTS.) 21 GO TO 2 22 S TtP E NI) C THIS FUNCTION SUBROUTINE EVALJATES TI-E DEFINITE INTEGRALS WHICH ARISE IN THE C ELEMENTS CF THE MATRIX T. FUNCTICN ALSOI(JPNSWri.F) EXTERNAL. FIPF2,F3tF4tF5 COMMON P1Iv T HETA 1=0.0THiErA2=P[/12.0.FLOATI,4P)) THETA3= PI-THETA2 OTHETA=2.0*THETA2 ALS%;I=C.0 00 30 IP=19MP ALSOJI=ALSQI. GAO SS6(THE TA 1,THE TA2, F) THE TA1=THETA2 30 THETA2=ThETA2+OTHErA AISQI=ALSC14GAOSS6ITHETA3#PI ,F) 9 RETUR4N END C Fl. F2v F39 F4 ANO F:5 AR~E AUXILIARY FUNCTION SUORCUTINES USED TO SIMPLIFY THE CINfEGIANUS OF THE DEFINITE INTEGRALS 14HICH ARE EVALUATED NUMERICALLY. FUNCTICN FI(THETA) COHMN P1 ,CASL-PSIMSEPMU#TSLPI)l4.NW2ONW2ENRW,-NSWHNSWH2.NW Fl=C0.5*C[S(FLI3AT(J)*flETA)-(Il.0)**J*SINTGL(THETASEPS1M,)J))*lSIN uiGLITHETASEPPOToNSWH) *l-1.O)**NSWH*SINTGL(THETASEPM1MNSWH) RETUORN END FUNCTICN F2ITIIETA) CuNI4UN P1 ,CASIEPSM9sEPMLTSEPMt~lNW2CNW2ENRWJNSWHNSWH2,NW F2=l0.54*CCS(FLUATIJ)*THETA)-l-1.0)**J*SINTCLIThETASEPSIM.JI )*(SIN I ;tL(1HEfAtS1:PPU1sN )W42) KET.JKN FNcrc 3(IE ENC F(IE CCHMON PI 9CA9 SEPS1I Ms!EPM~oTtSE PPil 1W2O NW2E NRW9 JtNSWH*NSI1Z, NW F3z1-l.03s*J.SINTGL(THETA.SEPSIM.J)*COSIFLOAT(NSWH)*THETA)s(-l.O)s k*NSwH*SINTGLI THETA. SiPSIMNSWI*CCSIFL0AT(J)STI-ETD)-2.0*C-I.0I**IJ L.NS)*SITGLTI-ETA.! EPSi1J)*SINTCL(THETASEPSI4,NSWH) R~ETURN END FUNL.TICN F4(I-htTA) CCMMCN PI.CA.SEPSli4.SEPMUTI.PPIM.NW2ONW2&.9NRWJNSWHtNWZNf F4=COSI2.O*FLCA1IJ)*THETA)*(ITGL(THETA, SEPML~tNSIEH).(-1.O)**NSWH L*SINTGL1THE TA*SEPMINWi)) RETUR' END FUNCTIEN F5(TIIETA)I COMM4ON PI.CA,SEPSIMtSEPHLTS EPP0IM.NW2CNW2LNRWJNSWHNSII2.NW NULL-0 F5-LER&C DO 4C gxl*NWZE SEPMU.TmCA*FLOAT (K-NKW1 SEPMZI4=CA*FLOAT INW*I-A-NRh) IF(K.EC.-W) GO TO 40 FSzF5*SINTtL(TI.ETAgSkPIU!,NULLI'-SINTGLITHETA.SEPIIMNULLI 40 CONTINUE SEP MLT=CA*FLOAT INWZO-NRWI IF(NW2O.NE.NW2E.AND.NRW.NE.NW20I FS51F5SISNTt;LITHETA.SEPMUTtNULL)+ ISINTGLITHETAtSEPSIMtNULL))*I0.5*CCS(FLOAT(JI*Tt-ETA)I-L-.0)**J*SINT 10L(1 TE1ASEPSIM9JI) [F(Nw20.EQ.Nw2E) FSIFS+S;4TGLTHETASEPSIMNULL))*(.5*COS(FLJAT( LJ)*THETA)-(-J.0)**J*SINTGLT1ETA, SEPSIMJ)I IF(NvZC.NE.NW2E.AND.NRw.EQ.NW2O) F5=F5*I0.5*CUSIFLUAT(J)*2.0*THETA I)) RET URN END C THIS FUNCTION SUBRO'UTINE EVALUATES 714E ANALYTIC EXPRESSION FOR THE DEFINITE C INTEGRAL SINTGL* FUNCTICN SINTGL(THETASEPIHAR)j COMMCN PI 50 IF (SEP) 51.55#52 5L PSI=ATAN(SIN(TF-ETA)/(.'2.0*.SEP-COS(THETA))I GO3 TG 53 52 PSI=PI-ATAN(SIN(THETA)/(2.0*SEP+COS(THETA))) 53 SUSQkT(4.C*ISEP)**241.044.0*SEP*CGSITHETA)) A*0 *5*CLS(ITHETA-PS I*FLUAT (AHAR-1)) Ba(l.042.0*SEP*COS(TrIE7A))*CUS(PSI*FLOAT(IHAR)ICi.*C(HY#SIFUIIA*) IF (IHAR.EQ.01 GO TO 54 SINTGL=(A*(S**2),B#S+C)/((i.O-S**2)*t-SI**(IHAR+1) GO TO 55 54 SINTGL=-IB*S*Z.0*C)/(S*(1.0-S**2) 55 RETURN END C GAUSS6 IS A FUNCTICN SUBUUUTINE W1IICH COMPUTES AN APPROXIMATE VALUE OF THE C INTEGRAL OF FIX) OV~ER THlE INTRbiVAL FRCP X=XL TO X=XU. EVALUATION IS i)ONE BY C MEANS OF A 6- POINT GAUSS QUAURATURE FORMUJLA. FUNCTICN GAUSS6(XLtXUF) A=. 5*( XU4XL I B=XU-XL Ca. 466234 8*13 uAUSS6x.oe566225*(F(A4C)4F(A-C) 4 C=. 330604 7*8 tAUSS6=GAUS56.1803808*(F(A+CJ+F(A-C)I C:. 1193096*8 uAUb So=b*(GALS~i6+.2339570*(F (A +C) F(A-C) I RETURN L. 1THI FUNCTION SUERCUTINt: SOLViS A SYSTEM OF N LINEAR EQUATIONS IN N UNKNOWNS L BIY USIN, GAUSSIAN lhLIM~INATICS will- COLUMN PIVOTING. LWAICAL FUNCTIOIN LSULVLIN) CCMMGN /tATRIX/A(2dv29I ilL ol 1=1,N 00 a! J~l,, )t SUM=SUt+ADS(A(IJJ) rUL~k= (SIJ!/FLCAT (NI**2 )*1 .JL-6 NMI=N-1 00 b6 K=1,NMI KP-K, 1 TEt4P=ABSIA(KtK)) t I TEMP=K 00 62 I=KP1,N I L (ABSIAI!,K)).LE.TkMP) GO TO 62 TEMP=AeS( A I K) I I TiMP I 62 CONTINUE IF- (rEMP.LE.TOLERI UJ TO 70 IF (ItFI4P.EQ.K) GO TO 64 00 63 I=K*NPI TEMP=A(K. I) A(K. I) A( ITEMP. I) b3 A(ITE?4P#I)=TE4P b4 O b5 I=i(P1,N A(It KI=A( I#KI/A(KK) 00 b5 J=KPlNPL 65 A(ItJ)=A(IvJl-A(ItK)*A('CJ) 66 CONTINUE IF(ABSCA(NN)).LE.1OLER) GO TO 70 AiNiNPI )A(NPNP1)IA(N#Nl i 00 67 J1I L=NPl-J 67 A(KNPI)=A(KNPI)-A(KtL)*AIL, NPl) 68 AIK#NPI)=A(KNP1)/AI. K) k69 LSOLVE=.FAL SE. GO TO 71t 70 LSOLVE%.TRUE. 71 RETURN END Appendix C Evaluation of the Integral I (03, mn-1, p) T1 I- ) I + Z(m -)(c/a) cosn -(cosr cos ' . it, -,p [ 4(m-') 2(c/a) + 2 + 4(m-L)(c/a) co,) + sin sine" I sin(pl')d'_ . (C-I) -(4(m-1)(c/a) + 2 coso ) cosa' -2 sino sino' ] The evaluation of I' (a, m-I, p) closely follows that for I(0, m-i, p) carried out in Appendix A. Using standard trigonometric identities the numerator and denominator of the integrand are reqritten as N 2 -A' cos [ (p-l)(ol' + i)] + B' cos [ p(a' + 4)] " C' cos [ (p+l)(n' + -El sin [ (p-1)(S' + # ]+ F1 sin [p(01 + )]"G' sin [(p+l)(o' + ] fJ D s +I +2s cos('+ +) (C-) where A' = sin(a -(p-1) ) (C-Za) B' -2(1 + 2(m-1)(c/a)coso) sin(p ) (C- 2b) C' -sin(O + (p+l)) (C-2c) E I cos(, -(p-1)) (C-2d) F' 2( + 21 cose) cos(pf) (C-2e) G' cos(a + (p+l) f) (C-Zf s s + + Zs cos(q' +i4) (C-3a) -106- -tan ( cos; (C- 3b) Z~m- Plc /a) + c 0 /.-I1 sina(a 'm-4(c/a) + Cose , (m-L) =-1, -2, -- The principle value of tan-I is used in (C-3b). With equations (C-2), (C-3) and the aew variable (=1' + s) (C-I) becomes rr I) -A cos[ (p-1) J] + B cos (p i, (o, n-,, p = - s4n Z ++Zs-cos( ) -Ccos[ (p-l) ] +E sin[p] -F sin[(p+)j]da' (C-4) The sin [ ( ) 4] terms integrate to zero and the remaining terms are in the form cf a definite integral which is evaluated in Appendix A, equation (A-7). With these integrations performed I' (0, m-I, p) is I' m, i-, p) = 1 [A's2 +B's +C'], p=l, 2-- (C-5) GI s 2)-)~ Appendix D Listing of Computer Program This appendix contains a computer program written in Fortran IV language for use on the I. B. M. 360/65 computer. The program computes the coefficients a and b ofmcp mcp i I the trigonometric series for the normalized correction current ti I densities gmc mc,, A C FORTRAN IV PROGRAM FOR CALCULATING THE CURRENT DISTRIBUTIONS CAUSED C BY INTERCON'iECTING WIRES IN THE EXPERIMERTAL MODEL. C C C THIS PROGRA4 USES THE METHOD OF COLLOCATION T3 SOLVE A SYSTEM OF C EQUATIONS FIR THE COEFFICIENTS OF TRIGONOMETRIC SERIES. THE SERIES C REPRESENT THE NORNALIZED SURFACE CURRENT DE4SITY ON EACH WIRE IN C SYSTEM OF NO WIRES. THE NUMBER OF HARMONIC TERMS USED TO DESCRIBE C THE CURRENT ON EACH WIRE IS tile THE RATIO CA IS EQUIVALENT TO THE C SPACING BETOEEN WIRE CENTERS DIVIDED BY THE WIRE DIAMETER. THE C NORMALIZED OIMENSIONS CAH2 AND CHL2 ARE THE TUBE RADIUS DIVIDED BY C THE TUBE HALF HEIGHT AND THE TUBE HALF HEIGHT DIVIDED BY THE LENGTH C 3F THE INTERCONNECTING WIRES* LOGICAL LINEGN COMMON THETASEPPI/MATRIX/A(36,37) WRITE1691) I FORMATIll) P1-3.141593 C AHZ ..48645 CHL 2=0.3*0824 2 REAO(5931 NWvNH*CA 3 FORMATfILs3X,12,3X#F5933 WRITE(6o4) NWvCANH 4 FORMATfI//35X11.,2IH WIRES, SPACING CA=,F6.3,AH, #12910H HARMON Iics) NW2-I2*NW-l+(-1 )**NWI/4 NSI ZEaNW21*NH NAU)G=NH*.'4W21 +1 00 17 LzLNH NRI= (NW21-11*NH+L C SETTING COLLOCATION P014TS THETA=P1S2.**FLOAT(L))/(FIOAT (NH.1lI 0O 17 M=L*NRINH NRW=1+ *Li/NH IF (NW2loEQ*NW2) GO TO 5 IFINRW.EJ.*NW211 THETAu(THETA-PI)12.0 IF(THETA*LE9090) THETA=2.D*Pt*THETA NIuNH 00 17 N=LNAUG IF (N.EQ.NAUG) GO TO 15 IF (N-NI) 7,7,6 6 NlaN1+NH NSW.NSW+1 7 NSIHaN-NHOINSW-11 IFf N&EQZ*(N/2)) NSHwNSH/2 IF(N*NE*2*(N/2)) NSH(NSN+11/2r IF (NRW*EQ.NSW) GO TO 11 SEPoCA*FLa3ATI NSW-NRW) IF i(2*NSWI*EQ*(NW,1)l GO TO 9 IFIZ*IN/Z)*EQ.N) GO TO B AIM N) --A INEt NSH) SEPaCA*FLUAT (NW~o--NSW-NRW) AIMNJ-I( MN)-(-1.O)**NSHWAINE(NSH) SEPmCAOFLOAT(INW+1-NSW-NRWIA(MN)'AI MN)4(-1.O)**NSH*&INES(NSHI r GO TO 179 IFIZ*(N/Z).EQelJ GO TO 10 AIN*N) z.%INE(2*NSH) GO TO 17 10 AIM9Nlu-AINES12*NSH-1) GO TO 17 11 IF I(2*NSw)*EQ.INW,1)l GO TJ 13I SEPaCA*FLOAT INmI*1-2*NRWIIF12*IN/Z)*EQ*N) GO TO 12AIN,Nlu(;OSIFLJAT(NSr4i*THETA) )/2.0-I-1.O)*NSH*AINEINSHI GO TO 17 12 A(MNI=I-SINIFLOATINSH)*THETA) I/2.*I-1.O)*eNSH*AINESINSH) £ GO TO 171,3 IF12*(N/2).EQoN) GO TO 14 AitNN)=t:JS(FLUAT(2*'4SH)OTHETA) 1/2.0 GO TO 17 14 A(NNiut$iN(FLOATI20'4SH-1)uTHETA))IZ.O (6i TO 17 15 SOURCE=0*O 00 16 l.01u1,NW SEPaCA*FLLJAT (LDI-NRWI SOURCI x (CAH2/S.IRTI1.3,(CHL2)**2,4.OS(SEP*CAH2*CHL2)**2))*((C LHL2)**2*(SINITHETAI-Z*O*SEPP:AH2*CHL2*COS(THETA))/(1*0+4.**SEP*CA &H20CHL2).,2)+SINITHETA)/I1.0O4*0.*(SEP*CAi42)**2))-I(1.O,2.osSEP*COS I(rHETA))/190.4.O*SEP**2,4,OPSEP*COS(THETA)))*I1.O-1.O/SQRTI1.O,4o LO*(CAH2)0*2*SEP**21) SOURCEavSJURCE+SJURC1 16 CONTINUE AIM, Nl=SJURCE 17 CONTINUE 18 IF (LINEJNINSIZE11 Gil TO 21 WRITEI69L9) 19 FORMAT(/*tUX31H'-THE HARMUNI; COEFFICIENTS ARE-) DO 22 LzLNWZ1 MsNH*( L-L I+L N-NH*L WRITE16,ZOI L 20 FORMATISIX6H WIRE 911) 21 FORMATISA,12F10.5) 22 CONTINUE THAMAIXSSIGLR N3UQU SOUINESS. GO TO 25 26 STOP END C THIS FUNCTI)N SUBROUTIRE EVALJATES THE ANkLYTIC EXPRESSION FOR THE C DEFINITE INTEGRAL AINE. FUNCTION AINE(IHAR) COMMON TAETAqSEP9PI/MATRIX/A136,37? 50 IFISEPI 31954952 51 IF(THETA.LE.PII PSIO-ATAN(SPilITHETAI/I.2.O*SEP.COS(THETA))) I F1THETA* 65.Pl) PSI=*ATAN(-SIN (THETA) /I -2,o 3SEP-COS(THETA) )I GO TO 53 52 IFITHETA.£LE.,PI) PSISPI.ATAN(SINITHETA)/12.O*SEPCOSITHETA)I) IIFITHETA.@GE*PI) PSImPI-ATANI-SIN(THETA)/I 2.O*SEPCOS(THETA))) 53 HuI1,O,.O*SEPSCOSITHETA))SCUS(PSI*FL2ATi 1MAR11 B.-0.5*CJSITHETA-PSIOFLOATI IHAR+1)) Cu-O.5*CJS(THETA+PSI;FLOATi IHAR-1l) Es4#.O*I 5-Pj S*2*2.O.4.O*SEP*CiJS(THETAI Gual (1O-F**2)**0.5-I*O)/F AINE=(G*,(IHAR-1))*(H*GB*(G**2),C2/lESI(1.0-F**2)**0.51) 54 RETURN END C THIS FUNCTIJN SUBRUUTINt EVALUATES WIE ANALYTIC EXPRESSION FOR THE C UEFINITE INTLGRAL AINES. FUNCTION AINESI IHAR) COMMON THETASEPPI/MATRIX/A(36,371 80 IF(SEP) slt84982 81 IF(THETA.LE.PI) PSI=-ATAN(SIN(THETA)/-2.*,EP-COS(THETAf) IF(THETA.GEoPI) PSI=ATAN(-SIN(THETA)/l-2.OD3tP-COS(THETA)I) GO TO 83 82 IF(THETA.LE.Pi) PSI=PI*ATAN(SIN(THETA)/(2.3*SEP4COS(THETA)I IFI rHETA.uE.PI) PSI=PI-ATANI-SIN(THETA)/(Z.O*SEP+COS(THETA))) 83 Hl=(1.O4.2.0*SEP*COS(THETA))PSIN(PSI*FLOATl1HAR)) 81=-O. 5*5INC PSI*FLJATI IHAR+L)-THETA) C1.-O.*5*SIN( PSIsFL3Art!HAR-1).THETAI E=4. O*lSIP)**Z+2.*4.0*SEPCJ3SITHETAI 4Fz(Z.0*k- 3..04*O.5)/E A 64 RETURN END C TF-IS FUNCTIJN SUBROUTINE SOLVES A SYSTEM OF N LINEAR EQUATIONS IN C N JNKNOWNS tsY US1'dG GAUSSIAN ELIMINATION WITrI COLUMN PIVOTING. LOGICAL FUNCTION LINEQN(N) CG%4MON TiETA#SEP9PI/MATRIX/AI36t37) 00 SUN=O.O DO 61 14.*N D0 61 J=L.N 61 SUM=SUM4A.SIA(IJi) i uLER ISUM/FLOAT(N)**2)*1.oE-6 NPI=N+l NMI=N-1 D0 66 Kzl*NM1 KPlzK41 TEMP=ABS At KK) I I TEMPzK 00 62 I=KP19N IF (ABS(A11,K)).LEeTEMP) GJ TO 62 TEMPmABSIA(IKI) ITEMPzI 62 CONTINUE IF (TE14P*LE*TOLER) Ga TO 70 4' IF (ITEMP*EQ*Kl GO TJ 64 00 63 !I(.9NPl TEMP=A(K.1I ACK, I)=AI ITEMP, II b3 AIITEMPti*llTEMP 6400O 65 Il'(P1,N All ,K)-Al I#)/AlKoK) DO 65 J=KP19NPI 66 CONTINUE IFIABS(A(N*N))*LE.TULER) GJ TO 70 A (N, NPI)zA(NoN P11/A (NN) D0 68 I=L#NMI KmN-I 00 67 Jul.I L=NP1-J 67 A(KNP1)-A(KNP13-A(KL)*AlLNPl) 68 A(KNPIAIKNP1)/A(KK) 69 LINEGNueFALSE* GO TO 71 70 LINEUN=*TRUE* 71 RETURN E-40 i I REFERENCES [1] R. W. P. King, Fundamental Electromagnetic Theory, Dover Publications, New York, Chapter 5, 1963. [2] S. Ramo, J. R. Whinnery and T. VanDuzer, Fields and Waves in Communication Electronics, John Wiley and Sons, New York, Chapter 5, 1965, pp. 286-303. [3] H.A. Wheeler, "Formulas for the Skin Effect", Proc. I.R.E., 30:412-424, 1942. [4] A. E. Kennelly, F.A. Laws, and P. H. Pierce, "Experimental Researches on Skin Effect in Conductors", Trans. A. I. E. E. 34: 1953-2018, 1915. [5] A. E. Kennelly, H.A. Affel, "Skin-Effect Resistance Measurements of Conductors to 100 Kilocycles", Proc. I. R. E., 4:523-574, 1916. [6] J. R. Carson, "Wave Propagation over Parallel Wires: The Proximity Effect", Phil. Mag., Ser. 6, 41:607-633, 1921. [7] H. B. Dwight, "Skin Effect and Proximity Effect in Tubular Conductors", Trans. A. I. E.E. , 41:189-198, 1922. [8] H. B. Dwight, "Proximity Effect in Wires and Thin Tubes", Trans. A. I. E. E., 42: 850-859, 19Z3. [9] S. Butterworth, "Eddy Current Losses in Cylindrical Conductors with Special Application to Alternating-Current Resistance of Short Coils", Phil. Trans. Roy. Soc. (London), Z22A: 57-100, 1921. [10] S. Butterworth, "On the Alternating Current Resistance of Solenoidal Coils", Proc. Roy. Soc. (London), 107: 693-715, 1925. [11] S. Butterworth, "Effective Resistance of Inductive Coils at Radio Frequencies", Experimental Wireless and Wireless Engineer, 3: 203-210, i091(itZ17-424, 483-492, 1926. [12] B. B. Austin, "Effectiv2 Resistance of Inductive Coils at Radio Frequency", Experimental Wireless and Wireless Engineer, 11: 12-16, 1934. [13] F. E. Terman, Radio Enginee:rs Handbook, McGraw Hill, New York, 1943, pp. 74-83. [14] E. C. Snelling, Soft Ferrites- .-Properties and Applications, C.R.C. Press, Cleveland, Ohio, Chapter 11, 1969. I5 -an. &G.. ~ r* -BET Fr~ce==3 3Ruistae and self capacvbm-cc sae s -Lajw.- Vf iclas:, Fraer. Z4.. ZSh 35-43. G. S. -cai. gmbe 74Xis= -=Mc!iaiC7 66L~Aab==- l Teea1 2= IV- Ea. Ccf SeCaLSvv"~se @t S ai-Bfz Ma4~ss .SmRgzIS @6r I At =- HLsTas AL Strg=i FB&Lvde, ofss. * CfW~~~ C~ti~erS~GaL New~ TC216 Pes. %232' ! ru - a,-po- E-45 1.Im Z -- Cj~m I- 3at&tCa am&i~~r P&M M~~as Cc-l.S-32 A~ z.. S.. (I~~a&ia) 75i1-T'. 35- i~ ~~~cz!, irae 3.. Mg~c- SrFt3O c Ma a ze*.. M.ra I$ I " new T' P.. r.d C.- .W * M~ T.?essx CI~s Masacseis I L 19, lm , 3 -113- [301 T. Padhi, 'Theory of Coil Anteans', Radio Science , 69D: 997-ICOI, 1965. [3J- G.A. Richards, 'ieaction Formulation and Numerical ResmIts for Multitorn L op Antennas and Arrays", Thesis, Ohio State Uaiversity. 1976. [32] R. W. P. King, Fundamental Electro.nag eic Theory, Dover Publizstions. New York, Chapter 6, 19.63. [33J R. W. P. King, The Theory of Linear Antennas, Harvard U-'u" ersity Press, Camkridge, Massachasettv, Chapter !Sec. 7. 1956. [34 D. Bierens DeHaan, Noawees Tables D'Integales Define3, L E. Stechert Co.. NeW Yo-k 7939, p. 89. [35" Jacke, Erode, and Losch. Tables of Higher Funcins. McGraw Hil, N e Yiork, 1960, p. L. [361 FA. Wheeler, "FandaznenUi Limitations of Srnali Antennas", L R.E. Proc., 35. 1479-1484. 1W-7. [371 H, Whiteside and W. P.P. Kin& o ba Loop Amtenza as a Pr-obeu. 1 L E. E. Transactious an Antennas and Propsgaia, bay 19K4 pp. 291-97. :I I