L I B RAR.Y 
 
 OF THE 
 
 UN IVLRSITY 
 
 OF ILLINOIS 
 
 6ZI.3G5 
 
 UGSStc 
 
 no. 2- 14 
 cop.3 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/inputimpedanceof06week 
 
INPUT IMPEDANCE 
 
 OF A SPHERICAL FERRITE ANTENNA 
 
 WITH A LATITUDINAL CURRENT 
 
 Contract No AF33(616) 310 
 RDO No B 112-110 SR-6f2 
 
 TECHNICAL REPORT NO. 6 
 
 by 
 
 W. L Weeks 
 Research Associate 
 
 20 August 1955 
 
 THE LIBRARY OF TllE 
 NOV 14 1955 
 
 UNIVERSITY Cf ILLINOIS 
 
 ANTENNA SECTION 
 
 ELECTRICAL ENGINEERING RESEARCH LABORATORY 
 
 ENGINEERING EXPERIMENT STATION 
 
 UNIVERSITY OF ILLINOIS 
 
 URBANA, ILLINOIS 
 
.8(p 
 
 TABLE OF CONTENTS 
 
 Page 
 
 3 tract iii 
 
 Acknowledgements Hi 
 
 List of Symbols iu 
 
 i Introduction 1 
 
 2. Field Equations 3 
 
 3 Solution of the Field Equations 4 
 
 4 Boundary Conditions and the Determination of the Coefficients 6 
 
 5. A General Expression for the Impedance 8 
 
 6, Application to a Ferrite Sphere 13 
 7- Examples of Particular Current Distributions 16 
 
 7.1 Sinusoidal Current Distribution 16 
 
 7.2 An Approximation to a Current Band 21 
 
 - i. I y Spherical Spheroid 26 
 
 
 iry and Conclusions 27 
 
 Appendix A - The Form of the Derivation for the Particular Case 
 
 of the Sphere 28 
 
 Appendix B - Some Details for the Problem of the Ferrite Spheroid 34 
 
 B~ 1 Prolate Spheroidal Coordinates 34 
 
 B 2 Oblate Spheroidal Coordinates 36 
 
 1 1 
 
ABSTRACT 
 
 An expression for the input impedance of a permeable spheroid with 
 
 a Latitudinal surface current is derived and evaluated for the case of a 
 
 il sphere. Formulas for the quality factor and radiation efficiency 
 
 are presented assuming simple current distributions. The extrapolation 
 
 of the results for the sphere for application to spheroids is considered. 
 
 ACKNOWLEDGEMENTS 
 
 It is a pleasure to acknowledge the suggestions and comments of 
 V. H. Rumsey and R. H. DuHamel, who also originally suggested the sphere 
 problem* 
 
 111 
 
GLOSSARY OF SYMBOLS 
 
 A auxiliary wave function (A = h r JE^) 
 
 D static demagnetization factor 
 
 E electric field intensity 
 
 radiation efficiency 
 1^ input current 
 
 I(n) =/dv -^L_ y n 
 
 h u h 
 
 J current density 
 n + Vi Bessel function of first kind 
 
 A 
 
 J n spherical Bessel function (See Eq. 46) 
 
 A 
 
 J n ' derivative of spherical Bessel function 
 
 K surface current density 
 
 K(0) K at equator (i.e., at z = 0) 
 
 A 
 
 spherical Hankel function (See Eq. 47) 
 a a 
 
 Y^ ' derivative of K_ 
 
 N + i/ Bessel function of second kind 
 
 P Legendre polynomial 
 
 P associated Legendre polynomial 
 
 equality factor - nL -^ 
 4 y Re z 
 
 B effective series resistance to represent ferrite losses 
 
 c 
 
 Rr> radiation resistance 
 
 U a solution of Eq 11 
 
 V a solution of Eq. 12 
 
 Z input impedance 
 
 IV 
 
GLOSSARY OF SYMBOLS (Cont. ) 
 
 a 
 
 
 ra 
 
 dius ( 
 
 ?f sphere 
 
 dS 
 
 
 element of surface 
 
 ds 
 
 
 element of arc 
 
 f (h u ,h v 
 g(v) 
 
 h ^) 
 
 = [h 
 
 h h^l'^ 
 " h v~p 
 
 J 1 U h 0_ 
 
 gi(v) 
 
 
 h h v 
 
 K 
 
 g2(v) 
 
 
 i 
 
 V h u 
 
 
 hv 
 
 h u ,h v ,hj ) metrical coefficients for the orthogonal coordinates u 
 v, and <fc, respectively 
 
 m 
 
 m 
 
 normalization factor for the functions V n 
 
 coefficients in the series expansion of the current 
 
 = w\l |ie = 2k/\ 
 
 dielectric constant 
 
 relative permeability (may be complex) 
 
 real component of K 
 
 imaginary component of K 
 
 wavelength of radiation 
 
 permeabi 1 1 ty 
 
 circular frequency 
 
1. INTRODUCTION 
 
 We will consider the problem of a non-conducting (but not lossless) 
 permeable sphere which is enclosed by a distribution of surface currents. 
 The currents are assumed: 
 
 a) to have no 4> variation 
 
 b) to have only a 4> direction, and 
 
 c) to be true surface currents, K, (in 
 the sense that 
 
 Figure 
 
 lim Jdr = K s 
 J-co dr-0 
 
 where J is the volume current density) 
 confined to the surface of a sphere, r = a. It is also assumed that the 
 generator of these currents does not disturb the symmetry. Under these 
 conditions, the input impedance can be found from the expression 
 
 Ui| 2 Z - - // E K* dS 
 
 in which I- is the input current, Z is the impedance, Ej> is the <fi 
 component of the electric field which is generated by the currents in 
 the presence of the ferrite and K is the surface current density. 
 The main problem is, therefore,, to determine Ej,. 
 
 The surface current distribution is an arbitrary function of 6. 
 For convenience, the arbitrary function will be represented by a series 
 of associated Legendre polynomials. 
 
 As a matter of fact, since it involves little extra difficulty, we 
 will not at first restrict the discussion to spheres. Instead we will 
 present a more general development which will apply to spheroids. Thus, 
 
we will discuss a problem in which the surface of the ferrite coincides 
 with some surface, u = constant, of a system of orthogonal coordinates 
 (u.v.0) in which the surfaces u = constant and v = constant are generated 
 by rotations about the z axis. The specific development for the case of 
 a sphere is presented in Appendix A as a detailed illustration. 
 
 The problem is admittedly an academic one. While the answer which 
 is obtained is correct if a current is established as specified, no 
 consideration is given here to the question of how such a current 
 distribution would be established. The results are nevertheless useful 
 as approximations to solutions for small antennas since in this case 
 the current should have negligible 4>- variation. .Furthermore, some 
 flexibility is retained in the formulation in that any 9 (or v) variation 
 can be represented. Thus it is possible to consider the problem of a 
 small ferrite sphere wrapped with a conducting ribbon at its equator, or 
 enclosed by the turns of a single layer winding of fine wire on its 
 surface. In the latter case, the surface density of current is inter- 
 preted as the number of turns per unit length of arc (meridian) times 
 the current in each turn. 
 
 2 
 
2. FIELD EQUATIONS 
 
 The electromagnetic fields must satisfy Maxwell's equations and the 
 boundary conditions. A form of the field equations applicable to this 
 problem (assuming e^ time convention) is 
 
 9(ML) a( h u Hj . u u ^ ... 3(h v E v ) achj^) . u u u 
 ~a^ t^ " jue h « h ^ (3> ~&r aT~ = ~ J ^ h » h »^ 
 
 wherein use has been made of the assumed symmetry to remove the deriva- 
 tives with respect to 0; h^h^h^ are the metrical coefficients for the 
 orthogonal coordinate system. Note that the equations consist of two 
 independent sets, one involving only H u , H v , and E^, and the other 
 involving only E u ,E y? and ILt,. We need concern ourselves only with the 
 first set (the left column above) since we require only E^ for the 
 impedance calculation. (As a matter of fact, it will be apparent from 
 the solution of the set on the left that the solution of the other set 
 which fits the boundary conditions gives Ey = E y = = PL,. ) 
 
 1. Schelkunoff, S. A. Advanced Antenna Theory, John Wiley & Sons, 1952. p. 4. 
 
3. SOLUTION OF THE FIELD EQUATIONS 
 
 An auxiliary wave function helps to solve the equations. We 
 introduce A, so defined that A = hJLy Then, 
 
 H, 
 
 H v 
 
 1 9A 
 jwu. h v h^ 9v 
 
 1 9A 
 
 jwn h^h u a u 
 
 (4) 
 
 (5) 
 
 E^ = A/h^ 
 
 The insertion of Eqs. 4, 5, and 6 into Eq. 3 gives 
 
 + 3. Jhi_9A 
 
 d_ (Nr_ 9A 
 3u \ h h u 9u 
 
 A substitution A = U(u) V(v) results in 
 
 9v (h h v 9vl | h 
 
 A ■ 0. 
 
 9_ 
 9u 
 
 h 
 
 I 
 
 h h u 
 
 v 9u + __ h v_. v 9fn + 9^ Al. y ay 
 
 9v h^h u 9u 2 9v (hyh^/ 9v 
 
 h v h 
 
 - u - u |4 + 
 
 U |J.8 
 
 H 
 
 UV - 
 
 (6) 
 
 (7) 
 
 If the variables are separable in the coordinate system selected then 
 the ratios of the coefficients in Eq. 7 must be such that the division 
 of the equation by UVf(u ; v) makes each term independent of either u or 
 v. A sufficient condition for the separation of the variables in Eq. 7 
 is then that the metrical coefficients satisfy 
 
 [g 2 (v)] (or more generally a [g 2 (v)g 4 (u)] ) (8) 
 V*u / 
 
 7~'^" fgi(u)] (or more generally ■ [gi(u)g 3 (v)] ) (9) 
 
 h v h 'A I 
 
NA 
 
 fi(u) + f 2 (v) 
 
 (10) 
 
 V> / 
 
 gi(u) g 2 (v) 
 
 in which the f ' s and g's are functions of the single variable indicated. 
 With these restrictions on the metrical coefficients, a multiplication 
 by gi(u) g 2 (v)/UV separates the variables and gives the two equations 
 (employing the less general conditions) 
 
 and 
 
 gi(u) U" + (u \ie fx(u) - k) U = 
 
 g 2 (v) V" + {k + u)V f 2 (v)} V - 
 
 (11) 
 
 (12) 
 
 where k is the separation constant. Thus, if the functions which 
 satisfy Eqs . 11 and 12 constitute complete orthogonal sets, the general 
 solution of the field equations (1, 2, and 3) is given by 
 
 dV, 
 
 jw^i h v h k k k dv 
 
 B, = —1 2 ak *k y k 
 ^ jwn h^h u k K du k 
 
 (13) 
 
 (14) 
 
 ^ = ij i a * Uk Vk 
 
 (15) 
 
 as may be shown by first writing down Eq„ 15 as the general solution for 
 E0, and then employing Eqs. 1 and 2 to obtain explicit expressions for 
 H u and Hy. 
 
 5- 
 
4, BOUNDARY CONDITIONS AND THE DETERMINATION OF THE COEFFICIENTS 
 The surface u - constant = u which coincides with the surface of 
 the ferrite divides space into two regions. Let the ferrite have 
 permeability |ii (may be complex) and the region outside u = u have 
 permeability u-2 (assumed real). On the surface u = u , the boundary 
 conditions on the tangential components of the fields require that 
 
 and 
 
 Lei 
 
 and 
 
 E 01 ( u o) = E^ 2 (u c ) (16) 
 
 H^ (u ) - H^ (u ) » K^ . (17) 
 
 Hi ■ ^ ^ a n U nl V n (18) 
 
 %2 ' t I b n U n2 V n («> 
 
 where U u - L as a solution of Eq. 11, which is finite at all points inside 
 
 the ferrite, and U n2 is a solution of Eq. 11 which represents outgoing 
 waves. Then 
 
 H vl = ■ * 2 a n d -^ V n (20) 
 
 JWU! h^ h u n " du n 
 
 "v2 = \r-T 2 b n -^ V n . (2D 
 
 jW|i 2 h^ h u n " du n 
 
 In view of the form of the expressions for the magnetic field, it will 
 
 be convenient to represent the v-variation of the surface current 
 distribution by a series as follows: 
 
 K, y ,(v) - K(0) f(h u , h v , h ) g(v) 2 a n V n . (22) 
 
The functions f and g are inserted at this point for convenience in a 
 later use of orthogonality properties. For the present, these functions 
 are arbitrary (real) except that they may have no zeros in the range. 
 K(0) is the value of the surface current density at the middle of the 
 range of v (i.e., at the equator, z = 0). The functions in Eq. 22 must 
 be such that the surface current density is finite everywhere and the 
 total input current (Eq. 36) is finite. The coefficients and the 
 functions V n are real. 
 
 The boundary conditions may now be employed to determine the 
 coefficients. From Eqs . 16, 18, and 19 we have 
 
 from which 
 
 2 a n "nl <"o) V n " 2 b n U n2 <«°> V r 
 
 - U n2 <"°> b 
 n "nl < u °> n 
 
 (23) 
 
 From Eqs. 17, 20, 21 and 22 we have 
 
 Z u i U n 2 ("o) ■ 
 
 2 ^ dU^Cuo) 
 
 jwki 2 h^ h u n n du junt h^ h u n n du 
 
 = K(0) f(h u , h v , h^) g(v) 2 a n V n . 
 
 (24) 
 
 Thus, putting dU n /du = U n ' we find, from Eqs. 23 and 24, that 
 
 = ju)^ 2 h^ h u K(0) f(h u , h v , h^) g( v) a n 
 
 U n2 '(u ) 
 
 V* U n2 ^ Uo ^ U n i'( u °) 
 Hi U n2 ' (u ) U nl (u ) 
 
 (25) 
 
 The value of b inserted in Eq 19 (or a n in Eq 18) gives E^, which is 
 required for the impedance calculation. 
 
 -7- 
 
5. A GENERAL EXPRESSION FOR THE IMPEDANCE 
 As was pointed out m the introduction, the impedance may be found 
 
 f 
 
 rom 
 
 lid* Z - ■ If E^(uo) \U* dS . (26) 
 
 surface a uo 
 
 Writing dS h y h^ dv dcp, and employing Eqs . 19 and 22, we find that 
 
 llJ'Z— // ± 2 b n U n2 (u ) V n K*(0) f(h u ,h v ,h ) g(v) 2 a m V m h v h^dvd0„ 
 
 * (27) 
 
 At this point we .specify the functions f and g, which were inserted into 
 
 the series expansion for the current, Note that if b n from Eq . 25 is 
 
 inserted explicitly into Eq.. 27, the metrical coefficients can be grouped 
 
 into a factor (h u h v h^) . Thus, since f (h u , h y , h^) appears squared in 
 
 that expression, a choir. 
 
 V l v ,h ) [h u h v h^] -K (28) 
 
 . :he integral of metrical coefficients. Next, we choose the 
 function g so that in the v integration in Eq. 27 we will have 
 
 / dv[g(v)] 2 V n V m (29) 
 
 range 
 and 
 
 / dv[g(v)] 2 V n 2 - A v (30) 
 
 range n 
 
 where Ay is a symbo] for the normalization factor, 
 
 ii 
 
 Provided that the variables separate in the manner indicated in 
 Bqa 8 ^ ;irul 10 (less general form), the function g can now be specified- 
 Foi suppose that two functions V^ and V_, represent two solutions to 
 
Eq. 12; then we have 
 
 and 
 
 g 2 (v) V k " + {k + wV f 2 (v)} V k = 
 
 g 2 (v) V n " + {n - wV f 2 (v)} V n - 
 
 If the first equation is multiplied by V n and the second by Vk 
 
 and subtracted from the first, we obtain 
 
 gs(v) 
 
 dv n dv 
 
 k dv / 
 
 (k n) V k V n - 0. 
 
 Dividing by g 2 (v) and integrating, we then have 
 
 fo A Vn 
 - v o ga(v) 
 
 k .. n |r. Vfdr-v? 1 -^^ 
 
 dv 
 
 dv 
 
 where the limits of integration are from pole to pole, so that if k / n, 
 
 we have 
 
 . v Vi V 
 
 (31) 
 
 j;;oYkA dv .. ( 
 
 o g 2 (v; 
 
 where g 2 (v) is the function which was specified in Eq. 8. Thus, if we 
 choose 
 
 g(v) = g 2 (v) /2 - 
 
 U> h u J 
 
 # 
 
 (32) 
 
 then Eq . 29 is satisfied since it is identical to Eq. 31. 
 
 When the values of f and g determined by Eqs. 28 and 32 are inserted 
 into Eq. 27, the final result is 
 
 iil 2 z - -2n 3 ^ 2 [K (o)] 2 r° 4. - ^ n * ^ m 
 
 U n ,'(Uo)_ H 2 U n /( U ) 
 
 n (33) 
 
 J n2 
 
 U n2 (u ) Hi U nl (u ) 
 
 2. Here we follow Schelkunoff, op. cit., p 115 
 
To save writing, we will define a function 
 
 f n (R,n) = 
 
 U n2 'U c) _ Ji2 U nl ; (Up) 
 
 U n2 (u ) Hi U nl (u ) 
 
 Note that the summation product indicated in Eq. 33 will consist of 
 
 terms of two sorts, viz: 
 
 a n a m Vn V m (a) 
 
 f n (R.Hi) 
 
 and 
 
 a 2 V 2 
 
 (b) 
 
 a V 
 n v n 
 
 f n (R, Hi) 
 
 By the orthogonality relation (Eq. 31), terms of the sort of (a) above 
 will give zero when integrated over the range of v. Utilizing the 
 definition (Eq. 30) for Ay , after interchanging the order of integration 
 and summation, Eq. 33 becomes 
 
 2 . 
 
 llJ'Z = -2tiju)u. 2 [K(0)] 2 2 - 
 
 n f n (R,M-i) 
 
 (35) 
 
 Next we relate the total input current to the surface current density 
 at the equator, K(0). The input current must equal the current which 
 crosses a half plane through the z axis of the ferrite. The incremental 
 contribution to this current is K(v) ds„ where ds is an element of width 
 along a v curve (that is, along the locus of the intersection of the 
 plane <£ constant, with the surface u ■ constant a u ). Thus, since 
 
 ds ■- h y dv 
 
 I, - / K(v) h dv 
 1 v 
 
 10- 
 
or, from Eqs . 22, 28 and 32, after interchanging the order of summation 
 and integration we have 
 
 I, ■ K(0) 2 a n /__ dv j-^- V n 
 
 (36) 
 
 range 
 
 l u n 4> 
 
 The integral in Eq, 36 is a definite integral (easily evaluated for the 
 case of a sphere, at least). If we define 
 
 I(n) ' dv r-V- V 
 then we have, from Eq„ 36, 
 
 range h u h^ n 
 
 (37) 
 
 Ii - K(0) 2 a n I(n) 
 
 l n n 
 
 (38) 
 
 Inserting Eq . 38 into Eq . 35 we have a general expression for the 
 impedance 
 
 a« Ai 
 
 -2ti jw|i 2 a n A V n 
 
 "[| a n I (n) ] r " fTrT^T 
 
 in which, by way of summary, the symbols are 
 
 a - L / dv K h V . 
 
 n K(0) A v "Lge ^ v n 
 v n 
 
 the coefficients in the series expansion of the current. 
 
 (39) 
 
 A v = / d 
 
 n range h u h^ n 
 
 I(n) = / dv —*— V n , 
 
 range h u h^ 
 
 £(R,Hi) 
 
 U n2 ? (up) _ ^2 U nl '(u ) 
 
 U n2 (u ) ^ U nl (u )_ 
 
 11 
 
The following form is usually more convenient: 
 
 -2k jwui 
 
 s "m <"°> 
 
 [f^ 1(b)] '"nl'W 
 
 2 
 
 a 
 n 
 
 Ay 
 n 
 
 
 
 E U n2 (u ) 
 
 
 (u ) 
 (u ) 
 
 - 1 
 
 _ 
 
 (40) 
 
 This expression is applicable to spheroids, or, in fact, to any shapes 
 which meet the symmetry requirements and the conditions on the metrical 
 coefficients . 
 
 12- 
 
6. APPLICATION TO A FERRITE SPHERE 
 
 We now return to the main problem of this report namely, to find 
 the input impedance of a ferrite sphere with latitudinal currents The 
 general expression (Eq 40) can be applied directly The angular and 
 radial functions are known to be Legendre polynomials and spherical 
 Eessel functions, respectively, (see Appendix A for details) The 
 computation of the other quantities in the expression is simplified if a 
 system of coordinates (r cos Q,4>) is selected, ^rather than the (r,8,^6) 
 system of Fig< 1 Thus , putting v : -cos 9, we find that the coordinates 
 (r,v,t£) are related to the rectangular system by the equations 
 
 x ~ r(l - v )* cos <p, 
 
 y = r(l - v )* sin (£, 
 
 z = -rv 
 
 so that the metrical coefficients are 
 
 K i 
 
 h y r/(l - v 2 f> 
 
 h "" rsJ l - v ' 
 
 (41) 
 
 as can be found from 
 
 h. 
 
 9>L] + \£X-\ 2 + -9z 
 a ^i 9 ^i 9 ^i 
 
 (42) 
 
 13 
 
rhere p.. is successively r ; v, and 4>) . Thus, since 
 
 V n = (1 - v 2 )^ P/ (v) (43) 
 
 for the sphere (P is the usual symbol for the associated Legendre 
 polynomial) we find 
 
 A v ■ J dv -Vv„ 2 = / dv — -J-rr [(1 - v 2 )^ P^ (v)] 2 
 h u h (1 - v ) 
 
 so that we can use a known result to find 
 
 A v - ill dv [Pn 1 ^)] 2 = la ir-H 1 ^ < 44 > 
 
 n 2n + 1 
 
 AJso 
 
 h 1 
 
 I (n) ' r -V V - / , dv (1 - v 2 )^ P/ (v) 
 
 
 P n (v) - (1 - V ) A — s , 
 
 dv 
 
 we have 
 
 I (n) = J*} dP n (v) = 2 . (45) 
 
 fhe appropriate radial function for the inside of the ferrite (r < a) is 
 
 J B (3 ir ) - 4np~r/2 J n + l/2 (3ir) (46) 
 
 ;irnJ for t.lif outside (r > a) is 
 
 K U3.r) ) "' 1 ' 1^772 [J n+H Or) j N n+H ((3r)]< (47) 
 
 14 
 
Thus for a sphere of ferrite of radius a the impedance (from Eq 40) is 
 
 z - 
 
 a 2 2n (n + II 
 _2n±_L 
 
 ^ TijhJii! J n (flta) 
 
 4 <f a J 2 " J n , ^i a )pT X'(j3 ia a) "" 3 n (3ia) 
 
 ^ 2 K (j3 2 a) ^ n '(3ia) 
 
 (48) 
 
 in which everything is specified except the coefficients of the current 
 expansion, a. 
 
 An explicit expression for the coefficients is obtained as follows 
 Let K(v) be a specified current distribution (for example. K(v) K(0), 
 -A < v <_ +A; K(v) = 0, otherwise). Then P by Eqs, 22, 28 and 32, 
 
 K(v) - K(0) [h u h^]" 1 I a n V n 
 or from Eqs. 41 and 43 
 
 K(V) = rd^V*)* 1 "n (1 -vV'P/U) 
 
 or 
 
 K(y) := Kmi s « n P - (v) 
 
 n n n 
 
 (49) 
 
 Thus, a multiplication of Eq 49 by P , and a subsequent use of the 
 orthogonality properties of the Legendre polynomials shows that the 
 coefficients are determined by 
 
 n K(0) 2n(n + 1) 
 
 ? n * ' * f$ dv K(v) P n J (v) 
 ntn + 1) 
 
 (49a) 
 
 Thus, Eq. 48 gives the input impedance in terms of completely specified 
 quantities . 
 
 15- 
 
7. EXAMPLES OF PARTICULAR CURRENT DISTRIRUTIONS 
 
 The expression (48) for the input impedance will now be evaluated 
 for two special cases. 
 7 i Sinusoidal Current Distribution 
 
 One of the simplest (computationwise) current distributions is one 
 in which 
 
 K(v) ■ K(0) (1 - v 2 )^ 2 = K(0) sin . 
 
 Such a distribution is of interest since it is known that a sinusoidal 
 distribution of direct current on a spherical surface gives a uniform 
 magnetic field inside the sphere. Hence, the results of the present 
 problem can be compared with the results of a familiar magnetos tatic 
 problem. Appropriate coefficients for such a distribution are a n = a, 
 
 1 and Ot. 
 
 0, n f 1. Thus, the series expression (48) for th< 
 
 impedance reduces to a single term as follows: 
 
 7 ... . Ji (Pia) 
 
 Z -- - 2/3 Ttjuu.! «— - — - 
 
 Ji (Pia) 
 
 |*i K 1 , (jP 2 a) o J, (P t a) _ 
 J* 2 K x (jp 2 a) Ji'(Pia) 
 
 (50) 
 
 This expression can be evaluated exactly by inserting 
 
 ^ sin 3ia * i [sin (3 3 a I 
 
 Ji(pa) ■ - cos 3^, Ji'((3a) = - A — - cos p x a + sin Pia 
 
 Pi a 1 Pxa / 
 
 Ki(jp 2 a) = e 
 
 j3 2 a 
 
 1 + ^— » Ki'(j(3 2 a) - -jP 2 Ki(jp 2 a) + 3 ~ 
 jP 2 a| P 2 a 
 
 so that 
 
 Ji'(Pia) 
 Ji Oia) 
 
 1 + 5i!f 
 
 1 - 3ia cot Pia 
 
 (51) 
 
 * H Static <md Dynamic Electricity, McGraw-Hill Book Co., Inc., New York, N Y. 
 
 2nd Edition, L950 p 274. 
 
 16- 
 
and 
 
 Ki'(jf3 2 a) 
 
 -JP2 
 
 (1 + j3 2 V) 
 
 (52) 
 
 Ri (j p 2a) a (3 2 a - j) a[l + (3 2 a) ] 
 
 The problem of interest at the present time is that of the small ferrite 
 
 A 
 
 antenna, that is 3a << 1 „ Writing the series expansion for Ji(3a)(or 
 Eq. 51) and retaining only the terms in the lowest order of (3a) gives 
 
 (53) 
 
 Similarly, from Eq„ 52 
 
 a j 
 
 (54) 
 
 Ki (j3 2 a) 
 for a non- radiating approximation. 
 
 Typically, the medium of Region 2 (outside the ferrite) will 
 be air, so that the ratio \i- : /\i z is m-/m- 2 " M^/m-o ~ K m .» the relative 
 permeability. 
 
 Substitution of these values in Eq. 50 gives, for a non radiating 
 
 approximation; 
 
 K 
 
 (55) 
 
 v ** K ■ m 
 
 The ratio of the impedance of a ferrite sphere to that of a sphere 
 
 having K - 1 is seen to ht 
 
 Zf 
 
 z 
 
 3K m 
 
 K m 
 
 (56) 
 
 Note that the limit of this ratio as K -• °° is equal to 3, and, at 
 K m - 100, it has the value 2,94 
 
 As was pointed out in Section 4 the permeability of the ferrite 
 
 may be complex, K = K ' - iK ".„ In this case Z has both a real and 
 
 ' r m m J m 
 
 17- 
 
imaginary component as follows: 
 
 Z - 2 niwu a <V - JV> 
 
 3 KjBll0,l iv + 2 ^ + 2) 
 
 (57) 
 
 f ^ 
 
 2 » V*- 
 
 m 
 
 If the resistance of the conductor is negligible (as has been assumed in 
 the derivation of the impedance),- $ quality factor may be found as 
 follows : 
 
 T (7\ K '(K ' 
 
 = lnnZJ - m v m 
 
 2) 
 
 Re(Z) 
 
 2K m ". 
 m 
 
 (58) 
 
 The radiation resistance may be calculated by carrying an additional 
 term in the approximation for the exterior radial function. We will, 
 therefore, app be Eq. 52 by 
 
 Ki'tj^a) . (1 + j(P a a) 
 
 Ki (j(3 2 a) 
 
 (59) 
 
 and Eq. 51 by 
 
 i 1 (Pia> = ft(l - 3/5(p ia ) 2 ) = ft . (60) 
 
 Ji'Oia) 
 
 The latter approximation is not consistent, but the factor (l-3/5((3i.a) ) 
 will not change the numerical results appreciably so it is omitted 
 to save algebra The insertion of Eqs ., 59 and 60 into Eq. 50 results in 
 
 Zi n ■ 
 
 K 
 
 (1 + j(P.a)*) 
 
 _m 2 
 
 (61) 
 
 18 
 
If K is real, a rearrangement gives 
 
 m 
 
 Z" 71 • 
 
 K. 
 
 J m " -2 J, 2 (P 2 a) 6 
 
 = | jW^ a 
 
 K 
 
 j -f~ «3 2 a) 
 
 L.4 + 1) *» ( ^ + 1) 
 
 The real part of this expression is the radiation resistance 
 
 R R -■- £ o)n a[K m / (-JVi)] 2 (P 2 a) 3 . 
 
 (62) 
 
 (63) 
 
 (64) 
 
 Note that the ratio of radiation resistance of a ferrite sphere to that 
 of a sphere having the permeability of free space is the same as the 
 square of the ratio (Eq« 56) The limit of the ratio as K -* °° is 9„ 
 When K = 100, the value of the ratio is 806= 
 If K is complex, we find, from Eq„ 61. 
 
 rr " K • 
 
 Z = £ jWHoa 
 
 K m" 
 
 m 
 
 jK_". 
 
 in 
 
 K ' K ' 
 LJ-O- + 1) (-*- + 1) 
 
 K' 3 
 j Y (3 2 a) 3 
 
 (V-.i) 2 
 
 wherein additional approximations have been made as follows 
 
 K 
 
 K_ 
 
 \ ' m 
 
 (65) 
 
 and 
 
 
 rn 
 
 Equation 65 shows two contributions to the real part of the impedance. 
 These have their origins in power dissipated in the ferrite and power 
 radiated, respectively. Thus a ferrite antenna radiation efficiency 
 
 19 
 
may be found as follows 
 
 Efficiency " 
 
 c + RR 1 + ( R c/Rr) ! + 
 
 2K m" 
 (V** ((B 2 a) 
 
 or 
 
 Efficiency - ■■ -^-7- 3n J (66) 
 
 m m 
 
 where V is the volume of the sphere, and R c is effective ferrite resist- 
 ance. 
 
 As was pointed out earlier, the assumption of a current distribution 
 K(v) = K(0) sin 9 allows the intermediate steps of the generalized 
 impedance calculation to be checked by a comparison with known results 
 of a magnetostatic problem. Employing Eqs. 13, 14, 23, 25, 28, 32, 53 
 and 54, one finds for the magnetic field inside the ferrite 
 
 H r = 2K(0) cos 9 
 
 (lit + 2\i 2 ) 
 
 H e = -2K(0) sin 9 
 
 (nx + 2|i 2 ) 
 
 Thus we find 
 
 H z ■ Ho - -^-- , where H ■■ 2&LQ1 
 
 |ii + 2n 2 3 
 
 which checks with the corresponding static solution. 
 
 In view of this agreement, it is tempting to express the results 
 in terms of the known static demagnetization factor for a sphere, D 
 1/3. If this is done, the results are in a convenient form for extrap- 
 olated application to other shapes for which the static demagnetization 
 factor is known (namely, ellipsoids and rods). Then, the ratio (Eq. 56) 
 
 20 
 
becomes 
 
 ?I = K m = K m ,, v 
 
 Zo 1+1(^-1) l + D S ( K m-D 
 
 and the limit of this ratio as K -* °° is 1/D g . In a similar fashion, 
 the quality factor is 
 
 Q . V'^A,"' (68) 
 
 vu - D s> + V <P= a > 3 
 
 \ O / 
 
 while the efficiency is 
 
 Km" 1 ~ D s X^ 
 
 In general, demagnetization factors are smaller for elongated shapes 
 
 than for flat stubby shapes. 
 
 7 2 An Approximation to a Current Band 
 
 The effect of a concentration of current (or windings) in the 
 equatorial region is studied by determining an appropriate set of current 
 coefficients.. A convenient set is a a = a, a 3 - - a/3? a 6 " .086a, and 
 ct = 0, n \ 1, 3, or 5. The current distribution which is determined by 
 these coefficients is plotted in Fig. 2. Once again, let us be content 
 with approximations for the radial functions which are valid for small 
 spheres (3a < .2). For this case, 
 
 jA - l xi'fk (eir)nTl (70) 
 
 and in the non- radiating approximation 
 
 K n - -j 2^^ ((3 2 r)" n (71) 
 
 2 n n! 
 
 21- 
 

 Figure 2 An Aonrox imat Ion to a Current Band 
 22- 
 
so that 
 
 J n (3ia) 
 J n '(Pia) 
 
 n + 1 
 
 (72) 
 
 and 
 
 K n '(jP 2 a) . n 
 A a • 
 
 K„ (j3 2 a) 
 
 (73) 
 
 Thus, Eq. 48 becomes 
 
 Z = — - — — r a 2. a. 
 
 2n 
 
 2 (.56) n n 2n + 1 
 
 
 
 1 
 
 
 
 
 f— 
 
 
 
 
 
 ■^ 
 
 M-i 
 
 
 n 
 
 
 + 
 
 1 
 
 Jig 
 
 n 
 
 + 
 
 1 
 
 
 
 (.56) 
 
 I — -EL— + 13 — JSn _ + 7 4 x in" 3 -5 -23 
 
 3 Krn + x 9 7 ( 3 XK m + 1) 11 (5/6^ + 1) 
 
 I— 2 
 
 (74) 
 
 The effect of the current concentration can be determined by comparing 
 
 Eqs. 74 and 55. (It is apparent that the absolute value of the impedance 
 
 is greater with an equatorial current concentration whatever the value 
 
 cf the permeability, but this is not the point of main interest here.) 
 
 The ratio of impedances with and without ferrite material (as in Eq. 56) 
 
 is 
 
 lim -£ - 2.92 , 
 K n"°° Z ° 
 
 as compared to the value of three with a sinusoidal current distribution. 
 The values of this ratio for other values of permeability are given in 
 Table 1. 
 
 Impedance Ratios Zr/Z (K m real) 
 
 Permeability 
 
 00 
 
 100 
 
 10 
 
 2 
 
 
 
 
 V Z o 
 
 % max 
 
 V Z o 
 
 % max 
 
 VZo 
 
 % max 
 
 Uniform Field 
 
 (sin 9 Distribution) 
 
 Approximation to Band 
 
 3.00 
 2.92 
 
 2.94 
 2.86 
 
 98 
 98- 
 
 2.5 
 2.47 
 
 83 
 84 
 
 1.5 
 1.51 
 
 50 
 52 
 
 Table I 
 
 -23- 
 
The effect on the quality factor can be assessed by allowing the 
 permeability in Eq. 74 to become complex. Table 2 gives the ratio 
 Q = Im Z'/Re Z for different permeability values. To make the results 
 comparable, -he loss tangent was assumed to be the same in each case 
 with the value tan 6 - V'/V = ° 01 - 
 
 Quality Factor for a Ferrite Sphere 
 
 Permeability 
 
 100 
 
 10 
 
 2 
 
 
 Q 
 
 ^sphere 
 ^toroid 
 
 Q 
 
 ^sphere 
 ^toroid 
 
 Q 
 
 ^sphere 
 ^toroid 
 
 Uniform Field 
 
 (sin 9 Distribution) 
 
 Approximation to Band 
 
 5100 
 5240 
 
 51 
 52.4 
 
 600 
 614 
 
 6. 
 6.14 
 
 200 
 207 
 
 2 
 2.07 
 
 Table 2 
 
 The ratio of the Q for a ferrite sphere to the corresponding Q for a 
 
 n Lte toroid with a current sheet winding is included for comparison. 
 
 averse is, of course, the ratio of the effective loss tangents.) 
 
 Note that the current concentration improves the Q slightly (copper loss 
 
 neglected) . 
 
 The effect of the concentration of current on the radiation resist- 
 ance is studied by replacing the radial function (Eq. 73) by a slightly 
 better approximation as follows: 
 
 K((JPi«) 
 
 r ' )3 ? a) 
 
 a 1 + ( p 2 a)^ '"• U +j(P2a) ] (?5) 
 
 : i). ! (3 + 2((3 a a) 2 + j(P 2 a) 3 ) ., l . 
 
 r 7~TZ — r a : 77 ri a L3 + j(p 2 a) J (76) 
 
 K, (j0 2 a) a 1 + (P 2 a) a 
 
 Ke(j6 a a) . ! (5 + 4(P 2 a) 2 + j(ft 2 a) 3 ) „ i .._ ... 
 
 K- ' |! . ;i ' 1 + (p 2 a) 
 
 24 
 
Employing Eqs. 75, 76, and 77 the expression for the impedance becomes 
 
 L i TCjw|i a 
 ; [.56] 
 
 1 !Sb + _3 5m 
 
 + 21 x io" 
 
 I 
 
 (5/61^ * 1) + j ^ (P 2 a) 3 
 
 Table 3 shows the very small effect on the radiation resistance. The 
 table exhibits the ratio of a ferrite filled spherical current,, Rnr to 
 the same current established on a sphere with permeability 1, Rp Q . 
 
 Permeability 
 
 100 
 
 10 
 
 Uniform Field 
 
 (sin 9 Distribution) 
 
 Approximation to Band 
 
 9 
 8,92 
 
 8.65 
 
 8.45 
 
 Table 3 
 
 6.25 
 6.14 
 
 2.25 
 2.24 
 
 RRf/^Ro 
 ffcf/RRp 
 
 ■25- 
 
8, THE NEARLY SPHERICAL SPHEROID 
 
 Equation 40 is valid for spheroids as well as for spheres; only the 
 functions are different. Furthermore, as is pointed out in Appendix B, 
 if the spheroids are nearly spherical, the spheroidal functions approach 
 those which correctly describe the sphere. Thus it is possible to 
 predict the results of calculations for nearly spherical spheroids. 
 Since the fields for small spheres check with those calculated for 
 corresponding static fields, we can expect that the static field solu 
 tions for spheroids in a uniform field will be equally valid. These 
 solutions are known and tabulated in terms of demagnetization factors. 
 The demagnetization factors for prolate spheroids, spheres and oblate 
 spheroids increase in that order. The impedance ratios expressed in 
 terms of the demagnetization factor are given in Eqs . 67, 68, and 69. 
 Note that, according to Eqs. 68 and 69, the Q and, to a much smaller 
 extent, the radiation efficiency increase with increasing demagnetization 
 factor (copper loss negligible). 
 
 
 \ (Men, J A Phys. Rev. 67. p. 351, 1945 
 
 -26- 
 
9. SUMMARY AND CONCLUSIONS 
 
 A formula for the input impedance of a permeable spheroid with a 
 latitudinal surface current has been derived and evaluated for the case 
 of a small sphere having two simple surface current distributions. The 
 variation of the magnitude of the surface current with angle (9) was 
 represented by a series of associated Legendre polynomials,. The effect 
 of concentrating the current in the equatorial region was considered; 
 there appears to be a small advantage in such a current concentration 
 (neglecting copper losses). Formulas for quality factor and radiation 
 efficiency are obtained. The results obtained for the problem of the 
 small sphere are extrapolated to the case of a small spheroid and the 
 advantage of flattened shapes is pointed out. 
 
 -27- 
 
APPENDIX A 
 
 THE FORM OF THE DERIVATION 
 FOR THE PARTICULAR CASE OF THE SPHERE 
 
 The general expression for the impedance which was derived in 
 Sections 2 to 5 of this report is applicable to antennas with shapes 
 which coincide with some surface of orthogonal coordinates with symmetry 
 about the z-axis The parallel argument for the particular case of a 
 sphere is presented here for the purpose of illustration. 
 
 In the spherical system of coordinates, the metrical coefficients 
 are h r = 1. \\q - r, h^ - r sin 0. Hence, the field equations (1, 2 and 
 3) become 
 
 3(r sin E^) . J(iHlr * sin o (A-l) 
 
 30 
 
 3 (r sin £^) = J(jJ)ir sin H (A 2) 
 
 30 
 
 3UHei _ 9^ . JW£rE (A „ 3) 
 
 3r 30 *' 
 
 To help in the solution of this system of equations, introduce 
 A = r sin E^. Then 
 
 H r - ■ . ,*. fl |f (A 4) 
 
 jwu-r sin 30 
 
 H Q 1 |A (A „ 5) 
 
 jwiir sin 3r 
 
 E^ ■ * . (A-6) 
 
 ^ r sin 
 
 2R 
 
The insertion of these equations into Eq. A-3 gives 
 
 3r 
 
 1 9 A 
 
 sin 9 3r 
 
 + _3_ I 1 3A + w_ngA _ q 
 39 r2 S in 9 39 sin 9 
 
 Put A = R(r) (9) to separate variables. Then 
 
 L_09!fi,l. _ 1 
 
 SI 
 
 n 9 3 r 39 
 
 r sin 
 
 R 30 
 
 R 39 
 
 + ] r 3_0 + oo 2 u,eR0 = o 
 
 r sin 9 39- sin9 
 
 (A-7) 
 
 Multiplication by r sin 9/R0 results in 
 
 rl 3jlR + s_in_9 3_ 1 
 
 R 3r 2 39 sin 
 
 39 
 
 (Note that in this case, the more general condition in Eq. 9 was satis- 
 fied so that gi(u) r and g 3 (v) sin 9. If we had employed the 
 coordinates (v, -cos 9, <p) as in Section 6, then the simpler condition 
 (Eq, 9), h u /h y hjj,, being a function of u only, would have obtained.) 
 The variables are now separated and we have the two equations. 
 
 R ;? + (w 2 ue -ii) R - 
 
 (A-ll) 
 
 and 
 
 sin 9 
 
 i__ 30 
 
 39 sin 9 39 
 
 ^ - k = 
 
 (A 12) 
 
 Equation A-ll is immediately recognized as a modified Eessel equation 
 for which the solutions are the spherical Bessel functions such as those 
 given in Eqs 46 and 47. A further transformation of Eq„ A 12 is 
 
 29 
 
helpful. If we introduce v = -cos 0, Eq A 12 becomes 
 
 (1 - v 2 ) |4 + k * . (A-12a) 
 
 ov 
 
 Then the substitution = (1 - v fi-V gives 
 
 ( i _ v 2 ) d!| _ 2v dY + k _ _1_J v ■ 0, 
 
 dv dv \ 1 - v / 
 
 which is immediately recognized as the associated Legendre equation 
 (with m - 1 and k = n(n + 1)). Hence V^ = P n (v) and 
 
 fr - V n - (1 - v 2 )^ P n 1 (v) - sin 9 P n X (-cos 9) 
 
 as was pointed out in Eq. 43 of this report. 
 
 Application of the boundary conditions (16 and 17) at the surface 
 r = a results in the equation 
 
 J a n J n (0,8) (1 v 2 )* P n X (v) - g b n K n (j3 2 a) (1 - v 2 )* P n X (v) 
 
 so that 
 
 A 
 
 _ K (j3 2 a) 
 
 'n-J^p7^) b " ■ (A " 23) 
 
 and the equation 
 
 1 . Q £ b n K n ' (j3 2 a) (1 - v 2 )* P ' (v) 
 jwu. 2 r sin 8 " " " n 
 
 1. 
 
 £ a n K n ' (j3 2 a) (1 v 2 )* P^ (v) 
 
 jwu- a r sin 
 K (0) f (h r h v h^) g (v) £ a n (1 - v 2 )* P n * (v) (A-24) 
 
 30 
 
so that 
 
 jw|i 2 r sin 6 K(0) f (h r , h y h^) g (v) <x n 
 
 'n a 
 
 V (j3 2 a) 
 
 1 
 
 H 2 
 
 K n (jr 2 a) %' ($ ia ) -, ' 
 
 Hi fc n (J3a a ) J n ((3i a ) 
 
 (A-25) 
 
 The impedance expression, Eq 27, is then 
 
 l± Z = 
 
 // 
 
 ^^j* ? b n K n (J3 2 a) (1 - v 2 )* 
 
 P^Ct) K(0) f (h r8 h v , h ) g (v) J a m V m r 2 dv d0 . (A 27) 
 
 Choose 
 
 f (h r? h v , h ) - [r*]-K 
 
 (A-28) 
 
 Choose g(v) so that 
 
 /_■* dv [ g (v)] 2 (i ~ v 2 ) p n i p m i = o 
 
 (A 29) 
 
 This suggests 
 
 g (v) - (1 - v 2 ) 
 
 ■X = 
 
 h h r 
 
 (A-32) 
 
 Then 
 
 I?} dv [g (v)] 2 (1 - v 2 ) (P,/) 2 - A v ■ 2n(n + I) 
 
 Y n 2n + 1 
 
 (A- 30) 
 
 Equation 35 for the case of a sphere is 
 
 V z ■ 
 
 2itjuu 2 [K(0)] 2 J 
 
 
 «ii 
 
 * 2n(n + 1) 
 
 (2n 
 
 - l) 
 
 T n '(j3 2 a)^ n 2 J n '(3 ia y 
 
 J^ (jp 2 a) [ x 1 J n (3 ia 2 
 
 (A 35) 
 
 31 
 
The input current in the case of a sphere of values a is 
 
 I, = f K(9) a ae. 
 
 i o 
 
 Since K is more conveniently expressed as a function of v rather than 
 the change of variables is incorporated here to give 
 
 h ■ J -i K <*> rf= 
 
 ,+ l v/„\ adv__ 
 
 V2 
 
 P 1 / \ 
 
 AK(0) 2 a n n - dv (A-36) 
 
 - 1 (i _ v '*)* 
 
 or 
 
 ,+1 P*(v) 
 h - K(0) 2 * n /_i ~ { f^^ ^ 
 
 = K(0) | a n f\ d P n (v) 
 
 ■ K(0) 2 I a n . (A 38) 
 
 Substitution of Eq . A 38 into Eq , A-35 gives Eq. 48 of Section 6 The 
 parallel argument is thus completed 
 
 If the problem of interest is that of a sphere with a number of 
 turns of fine wire in series (all carrying the same current) then 
 
 where n is the number of turns per unit length of arc (meridian) and Ij 
 is the current in one turn Thus we have 
 
 / K ds / n l { ds - l£ / nds - J X N 
 
 32 
 
so that the quantity 1^ in the main part of this report is equal to 
 /Kds/N Therefore each term in the expression for the impedance is 
 multiplied by N , where N is the total number of turns. 
 
 ■33- 
 
APPENDIX B 
 SOME DETAILS FOR THE PROBLEM OF THE FERRITE SPHEROID 
 
 The general expression for the impedance, Eq . 40, is valid for 
 spheroidal coordinate systems. 
 B I Prolate Spheroid 
 
 Figure 3 
 
 Let us consider the prolate spheroidal coordinate system in which 
 the coordinate surfaces u = constant and v = constant represent, respec 
 tively, families of confocal prolate spheroids and hyperboloids of two 
 sheets The numbers u and v are, respectively, the reciprocals of the 
 eccentricities of the generating ellipses and hyperbolas.. The metrical 
 coefficients are 
 
 / 
 
 2 2 
 
 u - 1 
 
 I 
 
 1- 
 
 h = i 4c? 
 
 1) (1 - V ) 
 
 where I ■ ae ' a/u is the distance from the center to the focus of the 
 ellipsoid, a is the semi- major axis, and e is the eccentricity. The 
 semi minor axis, b, is given by b ■ I nIu - 1. 
 
 The differential equations satisfied by U and V are found from 
 
 8, 9, 10, 11, and 12: 
 
 ga(v) 1(1 v 2 ) (B-8) 
 
 34- 
 
so 
 
 and 
 
 gl (u) = Uu 2 - 1) (B-9) 
 
 fi(u) = I 3 u 2 , f 2 (v) = -I 3 v 2 (B-10) 
 
 (u 2 - 1) U" + (wV I 2 u 2 - k) U = (B 11) 
 
 (1 - v 2 ) V" + (k - w 2 ne I 2 v 2 ) V = . (B-12) 
 
 It appears, therefore, that the "radial 88 functions U and the "angular 88 
 functions V satisfy the same differential equation. The substitution 
 V = (1 - v )' V transforms Eq. B= 12 into a differential equation which 
 
 2 2 
 
 becomes the associated Legendre equation as 3 I -* 0. The functions 
 which satisfy these equations are known and partially tabulated. 5,6 
 The other quantities in the impedance expression are: 
 
 Ay = \f l dv V = l/ 1 dv v~ 2 
 
 where 
 
 V n Ml - v 2 ) Y * V n , 
 
 and 
 
 Kn) - I/ 1 dv ^ 
 
 I -1 (1 - v ) 
 
 The current distribution is represented by a series as follows (Eqs. 22, 
 28, and 32) 
 
 l| t) ^ K(Q) | q n v n _ K (0) § a n v 
 
 r TTuf - vi (i - o z 2 ( u 2 - v 2 ) 
 
 % 
 
 5. Stratton, Morse, Chu and Hutner Elliptic Cylinder and Spherical Wave Func tions , The 
 Technology Press, Massachusetts Institute of Technology, in conjunction with John Wiley 
 & Sons, Inc., New York- 1941 
 
 6. Morse, P. M- and Feshbach, H Methods of Theoretical Physics, McGraw-Hill Book Co., Inc., 
 New York, New York, p 1502 ff. and pp. 1576-1579. 
 
 -35 
 
so that the coefficients may be determined from 
 
 K(0) A v 
 
 *\% 
 
 f dv (u 2 - v 2 ) /2 K^(v) V n 
 
 Thus, a formal procedure is prescribed for determining the input imped- 
 ance of a ferrite spheroid with a surface which coincides with the 
 prolate spheroidal coordinate surface u = constant. 
 B.2 Oblate Spheroidal Coordinates 
 
 Let us consider the oblate spheroidal coordinate system in which 
 the coordinate surfaces u = constant and v - constant represent, respec- 
 tively, families of confocal oblate spheroids and hyperboloids of one 
 sheet The numbers u and v are the reciprocals of the eccentricities of 
 the generating ellipses and hyperbolas. The metrical coefficients are 
 
 12 2W 
 
 u + v V 2 , 
 u 2 Tr ■ h < 
 
 1 
 
 \ 
 
 1 
 
 2 Va 
 
 v y i 
 
 —? . n 
 
 - v 
 
 xf> 
 
 I [(u 2 + 1) (1 - v 2 )] 1/z 
 
 where again I is the distance from the center of the spheroid to its 
 focus. The differential equations for U and V are found from Eqs ■ 8, 9, 
 10, 11, and 12 
 
 (u 2 + 1) U" + (wV I 2 u 2 - k) U - 
 
 (1 v 2 ) V" + (k + U)V I 2 v 2 ) V -■ 
 
 The transformation u : ju makes the equations for U and V identical. 
 
 2 V — 
 The substitution V (1 - v ) /2 V transforms the "angular" equation into 
 
 ;in equation for V which approaches the associate Legendre equation as 
 
 ' 0. The rest of the analysis closely parallels that of the pro- 
 
 Late spheroid Only thr; functions are different.. 
 
 36-