L I B R.AR.Y 
 
 OF THE 
 
 U N IVER.5ITY 
 
 Of ILLINOIS 
 
 6Z1365 
 
 IZ655te 
 
 no. 2- 14 
 cop-3 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/effectivepermeab09scot 
 
Antenna Laboratory 
 
 Technical Report No. 9 
 
 EFFECTIVE PERMEABILITY OF SPHEROIDAL SHELLS 
 
 by 
 
 E. J, Scott 
 
 and 
 
 R. H. DuHamel 
 
 15 April 1956 
 
 Contract AF33(616) -3220 
 Project No. 6(7 4600)Task 40572 
 
 WRIGHT AIR DEVELOPMENT CENTER 
 
 Electrical Engineering Research Laboratory 
 
 Engineering Experiment Station 
 
 University of Illinois 
 
 Urbana, Illinois 
 
J3& 
 
 ABSTRACT 
 
 The static field solutions for a spheroidal shell placed in a 
 uniform magnetic field are derived The solutions are then used to 
 calculate the effective permeability of prolate and oblate spheroidal 
 shells The results, in the form of sets of curves showing the 
 variation of the effective permeability with the shell thickness and 
 the ratio of major to minor axis, are applied in the comparison of 
 various types of cores for ferrite loop antenna applications. 
 
 1 1 
 
CONTENTS 
 
 Page 
 
 Abstract £i 
 
 1. Introduction 1 
 
 2. The Prolate Spheroidal Shell 3 
 
 3. The Oblate Spheroidal Shell 17 
 
 4. Application to Loop Antenna Design 32 
 
 5. Conclusions 39 
 6„ Acknowledgment 40 
 
 iii 
 
ILLUSTRATIONS 
 
 Figure 
 
 Number Page 
 
 1. Uniform Magnetic Field Applied on a Prolate Spheroidal Shell 4 
 
 2. Prolate Spheroidal Coordinates 4 
 
 3a. M- e versus [i for a Solid Prolate Spheroid (b/b = 0) for Various 
 
 Length-to-Diameter Ratios 10 
 
 3b. (i e versus \i for a Prolate Spheroidal Shell (b/b = .5) for 
 
 Various Length- to-Diameter Ratios 11 
 
 3c. [x e versus \1 for a Prolate Spheroidal Shell (b/b = .8) for 
 
 Various Length- to-Diameter Ratios 12 
 
 3d. \i e versus \1 for a Prolate Spheroidal Shell (b/b - .9) for 
 
 Various Length- to-Diameter Ratios 13 
 
 4a„ |i e versus b/b for a Prolate Spheroidal Shell (c/b = 1,25) for 
 
 Various \x 14 
 
 4b. \i Q versus b/b for a Prolate Spheroidal Shell (c/b = 2) for 
 
 Various (i 14 
 
 4c. |i e versus b/b for a Prolate Spheroidal Shell (c/b = 5) for 
 
 Various |i 15 
 
 4d. u. versus b/b for a Prolate Spheroidal Shell (c/b = 10) for 
 
 Various u. 16 
 
 5. Uniform Magnetic Field Applied to an Oblate Spheroidal Shell 
 with an Elliptical Loop Placed in the xz Plane 18 
 
 6, Oblate Spheroidal Coordinates 18 
 
 7a. [i versus |i for a Solid Oblate Spheroid (a/a = 0) for Various 
 
 Diameter- to-Thickness Ratios 22 
 
 7b„ |i versus |i for an Oblate Spheroidal Shell (a/a = .5) for 
 
 Various Diameter-to-Thickness Ratios 23 
 
 7c. [i versus \i for an Oblate Spheroidal Shell (a/a - .8) for 
 
 Various Diameter-to-Thickness Ratios 24 
 
 7d. |i versus |i for an Oblate Spheroidal Shell (a/a = .9) for 
 
 Various Diameter-to-Thickness Ratios 25 
 
 8a„ \i e versus a/a for an Oblate Spheroidal Shell (c/a = 1.25) for 
 
 Various |i 26 
 
 8b. |i versus a/a for an Oblate Spheroidal Shell (c/a = 2) for 
 
 Various \i 26 
 
 8c. p. versus a/a for an Oblate Spheroidal Shell (c/a = 5) for 
 
 Various |i 27 
 
 xv 
 
ILLUSTRATIONS (Cont. ) 
 
 Figure 
 
 Number Page 
 
 8cL |i versus a/a for an Oblate Spheroidal Shell (c/a - 10) for 
 
 Various |i 28 
 
 9o Uniform Magnetic Field Applied to an Oblate Spheroidal Shell 
 
 with a Circular Loop Placed in the yz Plane 29 
 
 10 |! e versus |i for a Solid Oblate Spheroid for Various Diameter- 
 
 to-Thickness Ratios (c/a) 31 
 
 11. |i e versus a/a for an Oblate Spheroidal Shell (c/a - 2) for 
 Various [i 31 
 
 12. Curves Showing the Length-to-Diameter Ratio, c/b, of a Solid 
 Prolate Spheroid Required to Produce the Same Effective 
 Permeability as a Solid Cylindrical Rod with a Length-to- 
 Diameter Ratio of L/D 32 
 
 13o Plot of (AmJ versus c/b for the Prolate Spheroid and c/a for 
 
 the Oblate Spheroid 35 
 
 14= Plot of (Am- ) versus c/b and c/a for the Solid Prolate and 
 
 Oblate Spheroidal Cores with c Held Constant 36- 
 
 15, Plot of (Ay, ) versus c/b for a Prolate Spheroidal Core with 
 
 c and the Core Volume Held Constant 38 
 
1. INTRODUCTION 
 
 For flush or nearly flush-mounted loop antennas a ferromagnetic 
 core is commonly used to increase the efficiency of the antenna The 
 recent development of low loss, high permeability ferrite materials has 
 made possible the design of ferrite loops with efficiencies equal to 
 those of much larger air core loops In the design of a ferrite core, 
 the engineer must decide on the shape, size, and weight, as well as the 
 type of material for the core. This work was undertaken to provide a 
 basis for the comparison of cores of various shapes, sizes and weights, 
 Since the radiation resistance of a ferrite loop is proportional to the 
 square of the effective permeability of the core and the loop area, the 
 efficiency of the loop is strongly dependent on these two quantities. 
 
 In the following, the results of the derivation and evaluation of 
 exact expressions for the effective permeability of prolate and oblate 
 spheroidal shells are given and are used to compare various types of 
 spheroidal shells Now, admittedly, the spheroid is not a practical 
 shape for a ferrite core However, two practical shapes, the cylin- 
 drical rod and the circular disk may be approximated by the prolate and 
 oblate spheroids, respectively The determination of the effective 
 permeability of a body involves the solution of a magnetostatic boundary 
 value problem (The static solution is accurate for bodies with dimen 
 sions which are small compared to the wavelength ) Exact solutions for 
 the rod, tube or disk would be prohibitively difficult to obtain since 
 the surfaces of these bodies do not coincide with a coordinate surface 
 for any of the coordinate systems for which Laplace s equation is 
 separable. On the other hand, spheroidal harmonics may be used to 
 obtain exact solutions for the spheroid or spheroidal shell Thus, 
 instead of obtaining approximate answers for the desired practical 
 shapes, we have obtained exact answers for approximate shapes 
 
 With regard to previous work an exact solution for the general 
 solid ellipsoid, of which the spheroid is a special case, was obtained 
 many years ago Extensive numerical results for the ellipsoid and 
 approximate results for the cylindrical rod have been given in recent 
 years i>2 In these references, the results are given in the form of a 
 demagnetization factor, D, from which the effective permeability, |i e , 
 
 1 Bozorth, R. M. and Chapin, D M. "Demagnetizing Factors of Rods," J App Phys,, 13, 
 (1942) p 320. 
 
 2 Osborn, J A "Demagnetizing Factors of the General Ellipsoid,"" Phys Rev., 67 (1945) 
 p. 351 
 
may be determined by 
 
 \K 
 
 e 1 + D(n 1) 
 
 where \i is the relative permeability of the core material For the 
 ellipsoid, D is independent of |i since a uniform field exists in an 
 ellipsoid when it is inserted into a uniform field.. However, since D 
 is a function of \i for an ellipsoidal or spheroidal shell, the advantage 
 of using D is lost„ Thus we will be concerned only with |i e . 
 
 2 
 
2. THE PROLATE SPHEROIDAL SHELL 
 
 The problem to be solved is defined as follows. With reference to 
 Fig. la, consider an applied uniform static magnetic field of intensity 
 Ho in the negative z direction which produces a total magnetic flux, 
 $o> through the circular wire loop of radius b„ 
 
 Next, consider the total flux, $ s , passing through the loop when a 
 prolate spheroidal shell of permeability u. and dimensions shown in 
 Fig. lb is placed inside the loop. The objective is to calculate the 
 ratio $ s /$ which is defined here as the effective permeability, |i e , of 
 the shell- 
 
 According to magnetostatic theory 
 
 H = -Vcp (1) 
 
 and 
 
 V 2 cp = (2) 
 
 where H is the total magnetic field and cp is a scalar function of posi- 
 tion. Thus the solution may be obtained by solving Laplace's equation, 
 subject to the appropriate boundary conditions. 
 
 It is convenient to use prolate spheroidal coordinates 3 (5,T],cp) in 
 the solution (see Fig 2). The outside surface of the shell is defined 
 by the prolate spheroid with semi axes b and c. The family of spheroids 
 which includes this surface is given by 
 
 c 2 (n -1) c 2 r\ 
 
 'here 
 
 p sin <p, y = p cos <p 
 
 2 2.2 
 C 2 = C D 
 
 l £ n < 
 
 00 
 
 3. Refer to W. R. Smythe, "Static and Dynamic Electricity," McGraw Hill, 1939; pp. 165-167. 
 

 Z 
 
 A "^ 
 
 
 1 if 
 
 Ix/l 
 
 1/ / 1 
 
 Y/A 
 7 /2 
 
 ~b'~ 
 
 --b 
 
 ^7 
 V 
 
 y 
 
 (a) (b) 
 
 Figure I Uniform Magnetic Field Applied on a Prolate Spheroidal Shell 
 
 £■1 
 
 — p 
 
 Figure 2. Prolate Spheroidal Coordinates 
 
 -4. 
 
The outside surface, T\ t> is defined by 
 
 TU 
 
 1 
 
 C2 h-(b/c) 
 
 Let the inside surface, r\ 2 , of the shell be defined by 
 
 n2 = {cl-J&l = ni >Il-K(b/c) 2 ', £ K < 1 
 
 c 2 
 
 It is then found that 
 
 b' 
 
 ^^^K 
 
 (4) 
 
 (5) 
 
 (6) 
 
 and 
 
 hi - b 
 ,.' c 
 
 JLK. 
 
 l-K(b/c) 
 
 (7) 
 
 The orthogonal family of coordinate surfaces are the hyperboloids 
 defined by 
 
 (8) 
 
 rhere 
 
 c 2 (l-£ ) c 2 £ 
 
 = 1 •: I ■»: 1 
 
 The cylindrical coordinates p and z are related to the prolate spheroi 
 dal coordinates <S and r\ by 
 
 Z = C 2 T|£ 
 
 P - C 2 [(1-£*)(t^-1)]^ 
 
 (9) 
 
 The prolate spheroidal harmonics which are solutions of Laplace's 
 equation are of the form 
 
 9= [A'P n m (£) + B / Q n m (S)] [AP n m (n) ^BQ n m (Ti)] x [ C cos mcp+D sin mf] (10) 
 
where P n m and Q n m are the associated Legendre functions of the first and 
 second kind, respectively. Since there is no Tj> variation for this 
 problem, m must equal zero. Also, since n (D = °°, the Q n (S) functions 
 cannot be used. 
 
 The potential corresponding to the applied uniform field is 
 
 q?o = H z = Ho r cos 9 (11) 
 
 where r and 9 are the spherical coordinates, Now as r\ -* °° 
 
 2 2 2 2 2 
 
 r = p + z -+ c 2 'n 
 
 so that r\ -• r/c 2 and £ = z/c 2 r\ -* z/r = cos 9, Thus £ can enter only as 
 Pi (5) = £ and the potential of the total field in Region I must be of 
 the form 
 
 cp s : = PA£)[APAn> *BQi(*)] 
 
 = a [An <-B(n coth" 1 !) 1)] (12) 
 
 = £• [A + BUoth" 1 *. 1)] . 
 
 Now, as z - 1 °°, t\ - °°, then r\ ' - and coth r\ w so that A may be 
 evaluated by equating Eqs „ 11 and 12 
 
 A - c 2 H o (13) 
 
 Therefore, for Region I, the potential is 
 
 cp s : ■ £[c 2 tt r\ + B(n coth" 1 n-Dl. (14) 
 
 In Region II between the prolate spheroids corresponding to 
 : t\i and r\ ■ r\ 2 the potential is of the form 
 
 cp s n ~- £[0\ +D(n coth -1 r\ 1)] (15) 
 
 end in Region III, since the potential must be finite, the potential 
 
 <p™ Ui\ (16) 
 
 -6- 
 
 1 1 
 
The four unknown constants, B, C, D, and E, may be evaluated by using 
 the following boundary conditions 
 
 T n 3qP J 3cp n 
 
 qpi = qp u j = ^ at 
 
 (17) 
 
 n m Qcpn 9q,m 
 
 9 * ' ^i*ri*r at ^^ 
 
 where (i is the relative permeability of the shell material. These 
 conditions yield four equations from which the four unknown constants 
 may be determined. The solution yields the following expressions for 
 the potentials: 
 
 ntCr*; i)^( l i-i)(cotK*r W .c ft tJf 1 ^ 1 ) + u - 1)nJ+1 (m)t ^ 
 
 r\i(r\ t 1) r\ 2 (r\ 2 2 ■ 1) 
 
 x (coth"^ I)D] 
 
 q>^ = lr\ [coth r\- k + - — .2 . , coth n 2 ]D 
 
 M (m. Dn 2 (Ti2 l) 
 
 (18) 
 
 
 (li l)r\ 2 (r\ 2 l) 
 
 where 
 
 c 2 H 
 
 D = 
 
 ,-i 1 r i (n i)n 2 + i 
 
 coth tu -=M -coth r\ 2 +- — r- t-^~TT 
 
 / ,-i l\ / a ,xf/ ,v'i .-1 u -i v^i(ti'l) + l n 2 (n-l)+l 
 
 coth ni-fTT hi^i" 1 ^^ D(coth n 2 coth tu)+ — : ^-p- - — n 77 
 
 V n V ] nt(rii -J.) r] 2 (,r\ 2 -1) 
 
 Now the flux passing through the loop may be obtained by inte 
 
 grating the normal flux density over the circle enclosed by the loop 
 
 The normal flux density is 
 
 r i 3qp 
 
 B £ = ^ H £ = ^ M- [Vcp] ^ - -u.-i- 
 
 c 2 
 
 r\ 35 
 
 £ = 
 
 -7 
 
The element of area on the equatorial plane is 
 
 da = C2TI dr\ dqp 
 
 Thus the total flux with the shell in position is 
 
 loop area 
 
 27i pru 
 
 B £ da 
 
 3cp s 
 
 ad, 
 
 S.-0 
 
 ■2kc. 
 
 3cpm 
 
 da 
 
 dr\ dqp 
 
 i\i 
 
 dr\ 
 «- 
 
 
 £ = 
 
 dn 
 
 Substituting and integrating gives 
 
 = -TCcsUDaru ■l)(cotH" 1 Ti 1 "Coth" 1 n 2 )-(ii 1 -T\ 2 ) 
 
 2 2,/ (u"l)n 2 + l 
 
 + (^i ri 2 ) I -, I +- 
 
 ,(m- l)n 2 (n 2 -l)/ (ui-i)r, a 
 
 For the shell removed, the total flux is 
 
 $0 = -H C2 Ho (tu -1) 
 
 Thus the effective permeability of the shell is 
 
 1 
 
 #0 
 
 ,-1 ,-1 (tu- n 2 )(Ti 2 ni + i) 
 
 coth n,! coth n, 2 +— - — 2 — — 2 — -+ 
 
 (n2-l)('0i - 1) (\i-l)Ai(nt 1). 
 
 (n Dn 2 + l 
 
 (coth TU-fr-J coth n 2 + - 
 
 (cotlTSi-f^) 
 
 (n Dn 2 (nM) 
 
 (coth 'rJTl^Tli 1)J(m. 
 
 1W ,-i ,-1 x ni(^-D + l n 2 (n 1)+1 
 
 l)(coth n, 2 coth il,)+ — ■ 5—77 — ; — 77 
 
 ni(T\l 1) llaC^a 1) 
 
 (19) 
 
 8 
 
It is helpful to know several limiting values of n_. Consider 
 first the case where b /b - 0, i.e., the effective permeability of a 
 solid prolate spheroid. As b'/b - 0, then b / /c / - 0, k - 1, and 
 r\ 2 - 1. By multiplying the numerator and denominator of Eq„ 19 by 
 (t\ 2 -1) and dropping the terms that approach zero, it is found that 
 for a solid cqre 
 
 lim ^e = u 
 
 b'/b - = ; ; — -3 
 
 l+(n-l)(ni-i)(nicotH" ru-l) 
 
 li 
 
 XHm 1) K.(§- coth 1 -?- 1) 
 
 C 2 ° 2 C 2 
 
 (20) 
 
 Next, consider the case where u. - °° It is easy to show that 
 
 lim ti e c 2 
 
 ^~*° b 2 (^- coth 1 g- 1) 
 
 (21) 
 
 Notice that for \l = °°, u e is independent of r> 2 and therefore independ 
 ent of the shell thickness. 
 
 Finally, let the length to diameter ratio increase without limit 
 
 (i.e„, let b/c - or T]i " 1) and keep b /b fixed. It is found that 
 
 lim [i a , 
 
 b/c - e = 1 + <H-U 
 
 1 (^) 2 .- (22) 
 
 The Illiac digital computer was used to calculate |i for various 
 values of the parameters c/b. b /b, and [A. Figures 3a to 3d illustrate 
 the variation of |i with [x for various length-to diameter ratios, c/b, 
 for four different shell thicknesses„ Figures 4a to 4d show the 
 variation of |i with the shell thickness for various permeabilities 
 and length-to-diameter ratios, It can readily be seen from these 
 curves that solid cores are necessary only for low permeability 
 materials. For example, notice that for a length to-diameter ratio 
 of five and a |i of 300 (which is not uncommon Ferramic J has a p, of 
 500 for instance) that |i is reduced only from 17 to 14 when approxi- 
 mately 81 percent (b /b = „9) of the core material is removed. 
 
 -9- 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 iih 1 ' 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r*b-H 
 
 1 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 (A \ 
 
 VA j 
 
 VKA c 
 
 11 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (j/i : 
 
 
 * 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^^^ m 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — io 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - — 
 
 
 
 
 
 
 
 
 
 
 : 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 §125 
 
 
 
 ^^^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Mr 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 rt 
 
 ] 
 
 )iam 
 
 ers 
 ete 
 
 1 < 
 
 us 
 r f 
 
 M f 
 iati 
 
 Dr < 
 
 DS 
 
 D 2 
 
 J Sol id P 
 
 O 3< 
 
 rolatc 
 
 ) 4< 
 
 1 Sp 
 10 
 
 D 5 
 
 ier 
 
 o e 
 oic 
 
 D 8 
 
 (b' 
 
 D l( 
 
 rb 
 
 X) 2( 
 
 0) for \ 
 
 X) 3C 
 
 i/ar ioi 
 
 XD 4C 
 
 is L< 
 
 O 5C 
 
 jng 
 
 o« 
 th- 
 
 X) »K 
 
 ■to 
 
o- 
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 0- 
 
 30 
 
 — 
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 0-- 
 
 — 
 
 20 
 
 15 
 
 10 
 
 0- 
 
 8- 
 
 4t5- 
 
 125 
 
 F 
 
 3 4 5 6 8 10 
 
 20 30 40 50 60 80 100 
 
 200 300 400 500 600 800 I00C 
 
 lure 3b n_ versus u, for a Prolate Spheroidal Shell (b/b = 5) for Various Length-to- 
 Diameter Ratios 
 
 11 
 
 lARY 
 UNIVERSITY OF HLIN0I9 
 
2 3 4 5 6 8 10 
 
 200 300 400 500 600 X) t, 
 
 3c - rerstll u for a Prolate Spheroidal Shell (b'/b 8) for Various Length-1- 
 Diameter Ratios 
 
 12 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 80, 
 '50 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "15 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "icf 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "if 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "T" 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ? 
 
 
 . 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I.S 
 
 
 
 
 
 c 
 
 '1.8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ir 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 F'ure 3d 
 
 > 2 
 
 Die 
 
 ver< 
 tmet< 
 
 \ e 
 
 JUS 
 
 ir 
 
 Rat 
 
 > e 
 
 for 
 ios 
 
 ii 
 a 
 
 3 2 
 
 Prolate S| 
 
 3 
 
 )heroi 
 
 4* 
 
 dal 
 13 
 
 3 5 
 
 Sh« 
 
 6 
 all 
 
 (b, 
 
 ) k 
 
 30 2( 
 
 9) for 
 
 X) 3 
 
 Var i( 
 
 00 4 
 MJS 
 
 DOS 
 
 Len 
 
 DOG 
 
 gth 
 
 co a 
 -to- 
 
 X) II 
 
 ;co 
 
Figure 4a u, e versus b'/b for a Prolate Spheroidal Shell 
 (c/b = I 25) for Various n 
 
 Me 
 
 
 
 
 
 /X=300 
 
 
 
 
 
 
 
 
 ~~ 
 
 ^iooN. \ 
 
 
 
 
 M« : 
 
 io^s. \ \ 
 
 
 
 
 
 /isioX^ 
 
 
 
 
 
 
 
 6 
 
 IX) 
 
 Figure lb , le versus b'/b for a Prolate Spheroidal Shell 
 (c/b 2) for Various ^i 
 L4 
 
Figure 4c* \s e versus b/b for a Prolate Spheroidal Shell 
 (c/b = 5) for Various \i 
 
 15- 
 
Figure Id. u,. versus b/b for a Prolate Spheroidal Shel 
 (c/b 10) for Various \.i 
 16 
 
3. THE OBLATE SPHEROIDAL SHELL 
 
 In obtaining the solution for the oblate spheroidal shell (Fig. 
 5a) a procedure similar to that for the prolate case is followed 
 Referring to Fig, 5b s let the semi axes a and c define the outer 
 surfaces j and a', c' the inner surfaces of an oblate spheroidal shell. 
 The family of oblate spheroids containing these surfaces is defined 
 by the equation 
 
 Ci C Ci(C +1) 
 
 where 
 
 y = p cos ~cp, z = p sin Ip 
 
 Ci = c 2 = a 2 , £ C £ co. 
 
 From Eq, 23 and Fig 5b it is seen that the value of C that de- 
 fines the outer surface is 
 
 Ci - a/ci - l/4(c/a) 2 -l . (24) 
 
 The inner surface can now be defined as 
 
 C 2 = k(a/cj = k£ x (25) 
 
 where £ k £ 1. 
 
 From the previous two definitions one obtains the relations 
 
 a'/a = k and c'/c = 4 1 f (k 2 -l )a7c* ." (26) 
 
 Corresponding to the spheroids are the family of orthogonal 
 hyperboloids given by the equations 
 
 P 2 
 
 _^, + * = lt (27) 
 
 ci£ ci(l-fi ) 
 
 where 1 ,: I <: 1 
 
 ■17- 
 
X 
 
 C j 
 
 
 
 
 c' m 
 
 
 
 
 
 
 *7/s 
 
 
 
 — r to 
 
 m 
 
 
 
 0' 
 
 
 
 
 
 y 
 
 (b) 
 
 Figure 5 Uniform Magnetic Field Applied to an Oblate Spheroidal Shell 
 with an Elliptical Loop Placed in the xz Plane 
 
 P 
 
 Figure 6 Oblate Spheroidal Coordinates 
 
 L8 
 
The oblate spheroidal coordinates of Fig- 6 are related to x and 
 
 P by 
 
 x = CiC£, 
 
 (28) 
 
 P = Cl [(K s )(l.^)] /2 . 
 
 For the problem under consideration the oblate spheroidal harmonics 
 are of the form 
 
 q? = [A / P a m (S) + B / Q n m (S)][AP n m (jO+BQ n m (jO} x [C cos m^D sin m?] . 
 
 First, consider the case of Fig, 5a when the applied uniform 
 magnetic field is in the negative y direction and an elliptical loop 
 lying in the xz plane is placed around the sphenoid., The potential of 
 the applied field is then 
 
 <Po = H y = H P cos <p „ 
 
 As £ . -+ co it may be shown that C ""• r/ci and £ -* cos 9 Also, as C -* °° 
 the potential with the spheroid present must approach the applied 
 potential, so 
 
 cp s = Hc Cl [(K 2 )(l £ 2 )] 1/2 cos cp . (29) 
 
 Thus it may be concluded from the form of the harmonic functions that 
 m = 1 and n = 1. The expressions for the associated Legendre functions 
 are 
 
 Q[(Z) - (l-£ 2 )% In l±f + -4r) 
 
 1~£ 1=5 
 
 p'(jO -j(K 2 ) 1/2 
 
 Since the potential is finite when £ = ± 1, Qi(£) must be excluded. 
 
 19- 
 
In Region I the potential is of the form 
 
 l= pf(£)[AP 1 / (jO^BQ 1 '( J a] cos v 
 
 = d-S^tAjd+C^+Bd+^Uot 1 ^--^-)] cos? 
 
 (30) 
 
 By letting C — °° it may be shown that 
 
 Aj = H ci 
 
 Therefore the potential in the three regions must be of the form 
 
 (Pg 1 = (l-fi^Al+C^^EHoCi+BCcot'C-^r)] cos <p 
 <pn = d-g^^d+^^EQj+DCcot'C-T^r)] cos? 
 
 cpj^ (1-S 2 )^(1+C 2 ) 1/2 Ej cos ? 
 
 The function Qi(jC) must be excluded from Region III in order for H to 
 be finite on the circle defined by t - 0, K, - 0., Applying the boundary 
 conditions at the surfaces Ci and C 2 the coefficients are determined to 
 be 
 
 -2mD 
 
 Ej d 
 Cj = -D 
 
 -1 2nd |i)C 2 
 
 COt L, 2 + 
 
 B D 
 
 K 
 
 (l-uJCad+Ca^ 
 
 ,. ,„ , -i, -1- N 2|i(l uKi Ci 2|i (1 ^)Cs 
 (1 li)Ci(cot Ci cot C 2 ) + — 75 • r-rs 
 
 1+Ci £2 i + c 2 
 
 and 
 
 I) Hoc >/ 
 
 C 2 l + £ 2 
 (cot Ci-cot C 2 ){1 + (1 M-)-^- - d -n)Ci— -Arot 1 ^} 
 
 2m (l n)Cl 
 
 cot Ci< M.- ( 1- U-) 
 
 C 
 
 I 
 
 + r— cot Ci 
 
 L+Ca ] (1 H)C 2 2r, 2 £ 2 2 
 
 20 
 
The flux density normal to the plane of the elliptical loop is 
 
 Bcp ■ UH, 
 
 evaluated at qp = ± k/2. The total flux passing through the loop is 
 
 i C 
 
 ' ^2 1 
 
 $ s = -4 Cl 
 
 
 HI 
 
 qp-j- 
 
 d£ dC 
 
 -4nc t 
 
 Ci o, 
 
 3qp 
 
 n 
 
 »2 
 
 £*+£* us. 
 
 dO(i.f) a<p 
 
 dS dC 
 
 After substituting and integrating there results 
 
 $ s - 7l^ Cl D 
 
 Ci(l + Ci) /2 (cot d-cot C 2 )'(l + Ci) /2 — 7- „f r M „a x v 
 
 C 2 (l-n)(l+C 2 ) (1 + dK 2 
 
 With the shell removed the total flux is 
 
 $o = rcabHc = rcHoc^CiCl^i 2 ) 172 
 
 The effective permeability for the oblate spheroidal shell is then 
 given by 
 
 ^e = ^ 
 
 cot Ci-cot C2 4 
 
 (ii"l)C 2 (K 2 ) d(l+Ci) 
 
 C 2 l+£ 2 
 
 (cot^l COt^jHl^lX'DyH^DCi-r- CoTCj 
 
 , / n x Ci 2 + Ci -t . ,x^i 
 
 y i+p cot ^Y+iv-DY 
 
 ITcl 1(n DC 2 2C 2 + C 2 2 cot 
 
 (31) 
 
 -21- 
 
, , , — 1 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 y-j 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 ^~~"^ 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 < 
 1 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 1 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ~y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 it 
 
 > 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 !T 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 S 
 
 
 
 
 
 
 iO 
 
 
 
 
 
 
 
 8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 
 
 
 
 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — — 1 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 fe 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^5^^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I I 
 
 1 < 
 
 I ! 
 
 > i 
 
 > 
 
 3 1 
 
 3 2 
 
 D J 
 
 41 
 
 D 5( 
 
 ) 6 
 
 & 
 
 D IC 
 
 2C 
 
 3( 
 
 X) 4C 
 
 K)5C 
 
 )0 € 
 
 00 XH 
 
 Figure 7a versus u. for a Solid Oblate Spheroid (a7a 0) for Various Diameter-to- 
 Tfiickness Ratios 
 
 ■22- 
 

 r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 03 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 50 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 30* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *2o" 
 
 IF" 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 
 
 
 8.6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 " 
 
 
 
 
 5.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 30 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2.0- 
 
 
 
 
 
 \ 
 
 :/o = 
 
 1.5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ' i 
 
 igure ' 
 
 2 ; 
 
 1 
 
 5 
 
 :o-TI 
 
 i ic 
 
 5 
 JS 
 
 <ne 
 
 5 
 
 li f( 
 ss f 
 
 6 
 
 )r < 
 {at 
 
 2 
 
 in Oblate 
 ios 
 
 3 
 
 Spher 
 
 4 
 
 oidc 
 
 to 5 
 
 il : 
 
 O 6 
 
 She 
 
 O 8 
 11 < 
 
 l( 
 
 )0 
 
 a - 
 
 2 
 
 5) f< 
 
 00 3 
 
 Dr Var 
 
 DO 4 
 
 iou; 
 
 DO 5 
 > D 
 
 006 
 
 ianr 
 
 .00 8C 
 
 eter 
 
 uia 
 
 23- 
 
1000 
 800 
 
 600 
 500 
 400 
 
 300 
 200 
 
 
 
 
 
 1 
 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 100 
 80 
 
 60 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 50 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 40 
 30 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 20 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Me 
 
 10 
 
 
 
 
 
 
 
 
 
 
 
 y 
 
 
 
 
 
 
 
 
 
 —"!s 
 — "k 
 
 (• 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — ■ ; 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 3 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -J 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 . 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Uji 
 
 3 4 5 6 8 10 
 
 200 300 4OO5006( fc*J 
 
 20 30 40 50 60 80 100 
 
 Figure 7c. u versus u. for an Oblate Spheroidal Shell (a'/a 8) for Various Diame>r-i 
 Thickness Ratios 
 
 24 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 00 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 50 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^s 
 
 
 
 
 30 
 15 
 
 To" 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r — T - 
 
 
 
 3.0 
 
 
 
 
 
 
 
 
 
 
 .^^ 
 
 
 
 
 
 
 
 
 2.0 
 
 
 
 
 
 
 
 :/a - 
 
 1.5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 4 5 6 8 10 
 
 200 300 400500600 800 I00O 
 
 20 30 40 50 60 80 100 
 
 hure 7d \x e versus u. for an Oblate Spheroidal Shell (a7a 9) for Various Diameter-to- 
 Thickness Ratios 
 
 25 
 
Figure 8a. u, e versus a'/a for an Oblate Spheroidal Shell 
 (c/a - I 25) for Various u. 
 
 Me 
 
 
 
 
 
 U=300 
 
 
 
 
 
 M^K. \ 
 
 
 
 
 
 /is ( 
 
 J^>v \ 1 
 
 
 
 
 
 
 
 
 
 
 
 1.0 
 
 Figure 8b. inversus a'/a for an Oblate Spheroidal Shell 
 (c/a 2) for Various |i 
 
 26^ 
 
Me 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 /JL =300 
 
 
 
 
 
 
 ' -^ 
 
 I - lOO^V 
 
 
 
 
 
 
 
 
 
 ^Sji^ 
 
 
 
 
 
 s^itsJO 
 
 
 
 
 
 
 
 
 
 
 
 
 .4 
 
 a 
 
 .6 
 
 .8 
 
 1.0 
 
 Figure 8c u e versus a'/a for an Oblate Spheroidal Shell 
 (c/a = 5) for Various \± 
 
 ■27- 
 
20 
 
 Me 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 k^iisiU 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 fl-\0 
 
 
 
 
 
 
 
 
 
 
 
 
 1.0 
 
 Figure 8d n e versus a'/a for an Oblate Spheroidal Shell 
 (c/a ■ 10) for Various u. 
 
 ■28- 
 
The limiting cases are given below For the solid core 
 
 lim [i 
 a'/a-0 
 
 l + (|i 1) -^ 
 
 /Ci 
 
 S- cot"" £- - -S- 
 
 Ci 
 
 Cl Ci 
 
 As |i — °° there results 
 
 lim p.. 
 
 oo 
 
 (ac'72ci)cot" 1 (a/c 1 )-(a72c'i) 
 
 Also, as c/a -* °° 8 we find that 
 
 lim |i e 
 
 c/a - oo = (1 k)|i+k . 
 
 Curves showing the variation of |i e with \i for the oblate 
 spheroidal shell are given in Figs 7a to 7d The variation of |i e 
 with the shell thickness is illustrated in Figs 8a to 8d 
 
 Next, consider the case of Fig 9 when the applied uniform 
 
 - z 
 
 Figure 9 Uniform Magnetic Field Applied to 
 an Oblate Spheroidal Shell with a 
 Circular Loop Placed in the yz Plane 
 
 29 
 
magnetic field is in the negative x direction and a circular loop 
 lying in the yz plane is placed around the spheroid. Since there is 
 no variation with cp for this case, the potential will take the 
 following general form: 
 
 cp = PA£)[APiW)+EQAjO]. 
 
 Following the same procedure as before, there results 
 
 C 2 £1 1 
 
 -1 o, -1 f, 
 
 cot Cl-COt C,2 + 
 
 "U2T ITCT " (ti-DCad + Ca) 
 
 (n-i)CM 
 
 cot Cj 
 
 (n-Dc: 2 (^+i) 
 +(Cicot 1 Ci-i)<|(M.-i)?:i-(ti-i)(?:i+i)(< 
 
 (c?+i>[(u-i)g £-i] 
 
 CaCSa+1) 
 
 For the solid core, there results 
 
 lim u., 
 a'/a-( 
 
 \i 
 
 1+(H-1) Sr (1 #- cot 1 £-) 
 Ci Ll < - 1 
 
 Also, for the shell, 
 
 lim |i e ci 
 
 M - 00 —7 
 
 ^ c 
 
 Ci 
 
 Cl 
 
 Figure 10 shows the variation of |i e versus |i for different 
 values of the ratio a/c in the case of a solid oblate spheroid The 
 variation of u. e versus shell thickness for a fixed ratio, c/a :: 2, 
 and different values of [i is illustrated in Fig. 11 
 
 30 
 

 
 
 N 
 
 
 if) 
 
 O 
 
 
 ■* 
 
 > 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 , 
 
 
 
 i 
 
 
 
 \ 
 
 A 
 
 
 
 \ 
 
 
 
 
 \\ 
 
 
 I 
 
 
 
 
 
 co 
 
 CO 
 
 CD 
 
 c 
 
 O 
 
 O 
 
 -M 
 
 l 
 
 i_ 
 o 
 +j 
 o 
 
 E 
 CO 
 
 O 
 
 CO 
 
 3 
 o 
 
 CO 
 
 
 "O 
 
 
 o 
 
 ^ 
 
 o 
 
 
 -C 
 
 
 ct 
 
 o 
 
 00 
 
 
 o 
 
 0> 
 
 -M 
 
 
 CO 
 
 CD 
 
 »-— 
 
 
 .ii 
 
 
 o 
 
 h- 
 
 
 
 T3 
 
 (0 
 
 _ 
 
 
 o 
 
 
 00 
 
 <? 
 
 :± CO 
 
 « 
 
 o 
 
 ^ 
 
 
 CO 
 
 
 L. 
 
 (O 
 
 CO 
 
 o 
 
 > 
 
 — 
 
 
 -+-< 
 
 o<o 
 
 =i ce 
 
 CD 
 
 i. 
 =3 
 
 
 7/8 
 
 r 
 
 /g 
 
 
 / 
 
 / ^ 
 
 
 / 
 
 
 
 
 
 1 
 
 
 
 CO 
 
 olo 
 
 IO 
 
 CD 
 
 JC 
 
 oo 
 
 1o 
 
 T3 
 
 O 
 Li 
 CD 
 
 JZ 
 Q. 
 OO 
 
 CD 
 -M 
 CO 
 
 X) zi 
 O 
 
 CO 
 
 C =3 
 
 CO O 
 
 t_ 1- 
 O CO 
 
 M- > 
 CO »- 
 
 CO H- 
 co 
 
 CO 
 
 L. II 
 CD 
 > CO 
 
 OO 
 
 o 
 
 L. 
 =3 
 
 en 
 
 CM 
 
 31 
 
4, APPLICATION TO LOOP ANTENNA DESIGN 
 
 The design of the winding for a ferrite loop antenna is just as 
 important as the design of the core However,, since there are various 
 factors to consider in the design of the winding, such as the type and 
 size of the conductor and the number and distribution of the turns, a 
 simple loop wound around the central cross section of the core has been 
 assumed in order to compare the different cores 
 
 The prolate spheroid is not an accurate approximation to the 
 cylindrical rod. A slightly larger length-to-diameter ratio is re- 
 quired for a prolate spheroid than for a rod of identical material in 
 order to have the same effective permeability. The curves of Fig 12 
 
 HJU 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 80 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 JL-K> 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 60 
 50 
 40 
 
 30 
 
 20 
 
 
 
 
 
 
 
 
 
 
 it -I 
 
 3 P 
 
 30 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 H- ' 
 
 I / 
 
 
 / y 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 5 
 4 
 
 3 
 2 
 
 l 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 6 
 
 8 10 
 
 k 
 D 
 
 20 
 
 ^0 40 50 60 80 100 
 
 Figure 12 Curves Showing the Length-to-D ameter Rat.o c/b, of a Solid 
 Prolate Spheroid Required to Produce the Same Effective Permeability as a 
 Solid Cylindrical Rod with a Length to-D, ameter Ratio of L/D 
 
 32- 
 
illustrate this for the solid prolate spheroid and for the cylindrical 
 rod Although the curves are not exact for shell type cores, they may 
 be used to improve the accuracy of estimating u. for a tubular cylin- 
 drical core That is, given a tube with a certain length-to-diameter 
 ratio, L/D, the corresponding length- to-diameter ratio, c/b, of the 
 spheroid would be read from Fig 12. Then, using the same shell 
 thickness, |i would be determined from Figs 3 or 4, 
 
 -« . ■ w « J- « *-"*-, ^v^ ^v-iinxilV^U i- L UMI A -L>5»' 
 
 The most important factor to consider in a loop design is the 
 efficiency or the signal-to-noise ratio which is proportional to the 
 efficiency The efficiency is defined by 
 
 Eff = i- 
 
 R in 
 
 where R r is the radiation resistance and R^ n is the input resistance 
 of the loop For the comparison of the cores, the important parameters 
 to consider are the [L & of the core and the area, A, of the loop, since 
 the radiation resistance is proportional to 
 
 R r - (A^x e ) 2 . 
 
 The input resistance is due to losses in the loop conductor and the 
 core, as well as to the radiation For electrically small loops the 
 radiation resistance is negligible compared to the conductor and 
 equivalent core resistance (Loop efficiencies of 10 are common) 
 The conductor losses are proportional to the length of the loop wire 
 and also depend to a lesser extent, but in a complex manner, upon the 
 geometry and construction of the winding and the core The core 
 losses also depend upon the type of winding and the shape of the core. 
 Thus it would be very difficult to use loop efficiency as a basis of 
 comparison of various cores For simplicity, then the square of the 
 factor A[X e will be used as a basis of comparison If the core loss is 
 negligible compared to the conductor loss (as it sometimes is for 
 ferrite loops at frequencies below 10 mc) then this is a good basis 
 for comparison because the conductor loss is, to the first order, 
 independent of the loop size and core shape, for a given loop in- 
 ductance, i.e., the loop efficiency is approximately proportional to 
 
 33- 
 
2 
 
 (Au ) On the other hand, if the conductor loss is negligible c 
 
 om- 
 
 pared to the core loss (as it may be for frequencies above 10 mc or 
 for some ferrite materials) this is a poor basis for comparison 
 because the core loss is approximately proportional to 
 
 R core °° H^Ad-l/Me) 
 
 Thus the efficiency would be nearly independent of the core shape, 
 especially for large n . Unfortunately, the core and conductor 
 losses are of the same order of magnitude for many cases so that it 
 is even more difficult to make theoretical comparisons 
 
 Consider first the case when the volume of a solid core is held 
 constant but the length-to-diameter or diameter- to- thickness ratio is 
 allowed to vary Curves of (Au. ) versus the ratio c/b for a prolate 
 spheroid and c/a for an oblate spheroid are shown in Fig 13. The 
 scales of c/b and c/a are arranged so that the loop area for prolate 
 and oblate spheroids is the same for any one abscissa However, since 
 the loop is assumed to be tightly wound around the spheroid and since 
 the volume of the core is held constant, the loop area decreases with 
 increasing abscissa Two important conclusions may be drawn from 
 these curves First, for a given volume of core material t <ere are 
 optimum values of the ratios c/b and c/a for the prolate and oblate 
 spheroids, respectively. The optimum value depends upon the permea 
 bility of the material For materials with a high permeability it may 
 not be practical to use the optimum ratio (for example, with U- 5 100, 
 a c/a ratio of 150 would be required) . Second, for a given volume of 
 core material the oblate shape is preferable since the value of (A|i e ) 
 is up to three times greater than that for the prolate shape 
 
 Next, consider the case for which the major axis of a solid core 
 is \,oA<i constant and the other parameters (the volume and minor axis) 
 are allowed to vary This condition would apply to a flush-mounted 
 ferrite Loop [.laced in a cylindrical cavity of fixed diameter Figure 
 U above (Au R ) 2 aa a function of c/b and c/a for the prolate and 
 Mate apheroida, respectively. For these curves the length of the 
 ! "''■'" spi.-M.id ;uid !.!.»■ 'h;imetex "I the oblate spheroid are held 
 
 i tti 
 
 rMttll La tO b« published shortly in an Antenna laboratory Report by W I, Weeks 
 [«1 t i » • I OOp anil Tinas 
 
 -34- 
 
< 
 
 
 
 
 
 Prolate 
 
 
 
 
 
 
 Oblate 
 
 
 
 
 
 ?no 
 
 Spheroid 
 
 
 Spheroid 
 
 
 
 i ' 
 
 
 
 
 
 4 
 
 ^ 
 
 
 
 
 
 
 
 
 m 
 
 
 
 
 
 
 
 / 
 
 / 
 
 r 
 
 /1=I00-t^ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 flO 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 f 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 cr> 
 
 
 
 
 
 r 
 / 
 
 
 
 
 
 
 ^ 
 
 < 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Af\ 
 
 
 
 
 / 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *tU 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 * ^ s 
 
 
 
 
 
 
 
 
 
 90 
 
 
 
 / 
 / 
 
 / 
 
 S y 
 
 i 
 i 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 > 
 
 '// 
 
 
 
 l 
 
 
 
 ^ -> 
 
 
 
 
 
 
 
 
 10 
 
 / 
 
 £ 
 
 rjr 
 
 
 
 1 
 | 
 
 
 
 ^ r30 _/-***> 
 
 \ 
 
 
 
 
 
 
 
 
 ^? 
 
 ► ^^^ 
 
 
 
 
 
 
 
 
 
 >v > 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 
 
 
 
 
 
 
 
 
 
 
 \\ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 6 10 
 
 C 
 
 F 
 
 20 
 
 16 36 64 100 £. 400 
 
 a 
 
 40 60 80 100 
 
 1600 3600 6400 10,000 
 
 Figure 13 
 
 Plot of (A;j, e )2 versus c/b for the Prolate Spheroid 
 and c/a for the Oblate Spheroid (The volume of the 
 core is held constant and the area of the loop for 
 the prolate and oblate spheroid is identical for 
 any one abscissa However, the loop area for both 
 varies with the abscissa ) 
 
 35- 
 
2C 
 
 X 
 
 Prolate 
 
 Top 
 View 
 
 Side 
 View 
 
 w 
 
 5 i 
 
 10 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 3 
 
 2 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■- 
 
 ? 
 
 
 -Oblate 
 c = constan 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^M : 
 
 100 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 fl =30 
 
 
 
 
 
 
 
 
 .6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Prolntft 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 c-constan 
 
 t 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 2 
 
 J 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 H- 
 
 ?\ 
 
 X) 
 
 
 
 
 
 
 
 
 
 
 
 fi: 
 
 \Q-* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6 8 10 
 c/b and c/a 
 
 20 
 
 30 40 
 
 60 80 100 
 
 Figure W 
 
 Plot of {k[x e r versus c/b and c/a for the Solid 
 Prolate and Oblate Spheroidal Cores with c Held 
 Constant 
 
 ■',(> 
 
constant. The volume of the core decreases with increasing abscissa. 
 It is seen that the spherical shape has the largest value of (Apt ) 
 This is to be expected since the loop area and volume of the core are 
 maximum for this case Also, it is apparent that when c/b and c/a are 
 equal, the oblate shape is better than the prolate shape. For example, 
 for c/a - c/b = 10 the value of (A|i ) for the oblate is 15 times that 
 for the prolate spheroid for u. = 100 This is due to the fact that the 
 loop area and core volume for the oblate core are ten times greater 
 than those for the prolate core. 
 
 Consider now a prolate core with constant length and volume of 
 material, but let the shell thickness change Figure 15 illustrates 
 the variation of (An e ) and the shell thickness as a function of the 
 length to-diameter ratio Again it will be noticed that as the 
 spherical shape is approached, (A|! e )~ increases rapidly, 
 
 37 
 
'.n-o 
 
 Figure 15 Plot of (Am, 6 ) 2 versus c/b for a Prolate Spheroidal 
 Core with c and the Core Volume Held Constant 
 
 38 
 
5. CONCLUSIONS 
 
 The sets of curves illustrating the variation of |i e versus the 
 shell thickness and length to diameter ratio for prolate and oblate 
 spheroidal shells are quite useful for comparing the effectiveness of 
 various types of cores, For airborne ferrite loop applications where 
 the length, height, and weight are limited, the following rules of 
 thumb may be applied. First, make the loop area as large as possible. 
 Next, use the maximum weight of core material allowed and distribute it 
 in a shell form which just fits inside of the loop* For rotating or 
 crossed loop applications this will usually result in an oblate 
 spheroidal shell type of core with an elliptical loop. 
 
 39 
 
6. ACKNOWLEDGMENT 
 
 It is a pleasure to acknowledge the work of Mr. Earl J, Schweppe 
 who evaluated the expressions for the effective permeability by means 
 of the Illiac digital computer. 
 
 40- 
 
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 Great Neck 
 
 Long Island., New York 
 
 M/F Contract AF33(038) -14524 
 
 Temco Aircraft Corp 
 
 Attn- Antenna Design Group 
 
 Dallas., Texas 
 
 M/F Contract AF33( 600) -31714