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 ANTENNA LABORATORY 
 Technical Report No. 13 
 
 IMPEDANCE OF 
 FERRITE LOOP ANTENNAS 
 
 by 
 
 V. H. Rumsey 
 
 and 
 
 W. L. Weeks 
 
 15 October 1956 
 
 Contract No. AF33(616)-3220 
 
 Project No. 6(7-4600) Task 40572 
 
 WRIGHT AIR DEVELOPMENT CENTER 
 
 This volume Is bound without 
 
 no. ih 
 
 which is/are unavailable, 
 
 >EARCH LABORATORY 
 lENT STATION 
 
 vrnu v tiwi I I \jr ILLINOIS 
 
 URBANA, ILLINOIS 
 
no. 13 
 
 
 ANTENNA LABORATORY 
 Technical Report No. 13 
 
 IMPEDANCE OF 
 FERRITE LOOP ANTENNAS 
 
 by 
 V. H. Rumsey 
 
 and 
 W. L. Weeks 
 
 15 October 1956 
 
 Contract No. 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 
 
Antenna Laboratory 
 Technical Report No. 13 
 
 IMPEDANCE OF FERRITE LOOP ANTENNAS 
 
 by 
 
 V.H. Rumsey 
 
 and 
 W.L. Weeks 
 
 15 October 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 
 
 UNIVERSITY OF 
 
 ILLINOIS LIBRARY 
 
 AT URBANA-CHAMPAIGN 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/impedanceofferri13rums 
 
ABSTRACT 
 
 A variational formula for the imput impedance of a loop antenna with 
 
 a ferrite core has been derived in terms of an assumed volume distribu 
 tion of magnetization in the core. It takes the form 
 
 s I 
 
 ///core i ' -?-- 7 ~ H k d V 
 
 core |jW(n - |i Q ) k j 
 
 Where Z = impedance of the antenna 
 
 Z Q = impedance when the ferrite is removed 
 
 2. ~ surface density of electric current on the winding, assumed 
 perfectly conducting 
 
 K = assumed volume density of magnetic dipoles 
 
 (J. and |i represent the permeability of the ferrite and free space 
 
 Ek and Hu represent the free space field due to K„ 
 
 Effects of the winding arrangement" core losses and shape of core 
 are contained in the formula. It reduces to a simple expression when 
 applied to an electrically small antenna with ellipsoidal core. 
 
CONTENTS 
 
 Page 
 
 Abstract ii 
 
 List of Symbols iv 
 
 1. Introduction 1 
 
 2. Formulas Based on an Assumed Distribution in the Core 4 
 
 3. Formulas in Terms of the Demagnetization Factor 7 
 
 References 14 
 
 Appendix I --Derivation of the Formula I^Z = ~//s J • E dS 15 
 
 Appendix II- -Derivation of Equation 11 19 
 
 Appendix III Stationary Properties 21 
 
 Appendix IV Calculation of Inductance Using Static Theory 23 
 
LIST OF SYMBOLS 
 
 A area 
 
 D demagnetization factor 
 
 E electric field intensity 
 
 E f E generated by ferrite magnetic dipoles in free space 
 
 E t component of E tangent to the path of the wire 
 
 E w E generated by wire currents in free space 
 
 H magnetic field intensity 
 
 H c component of H in the core which is normal to plane of loop 
 
 Hf H generated by ferrite magnetic dipoles in free space 
 
 H Q magnetic field of plane wave incident on loop 
 
 1^ magnetic field generated by wire currents in free space 
 
 I electric current 
 
 J surface density of electric current 
 
 K density of induced magnetic dipoles 
 
 K a assumed (or approximate) distribution of K 
 
 N unit vector normal to area 
 
 to the order of 
 
 P refers to point at source of field 
 
 Q refers to point of observation 
 
 R resistance 
 
 Rp radiation resistance 
 
 S general surface 
 
 V volume 
 
 U input voltage 
 
 Z input impedance of antenna 
 
 Z Q Z of antenna in free space 
 
 Z Qe Z Q for ellipsoidal antenna winding 
 
 ^oL ^o ^ or s i- n §l e turn loop 
 
 a length of ellipsoid semi-axis (2a is length of core measured along 
 the length of the winding) 
 
 a p symbolic representation for the source consisting of the assumed (or 
 approximate) distribution of magnetic dipoles in the ferrite 
 
 b ellipsoid semi-axis, or radius of circular loop around the core 
 
LIST OF SYMBOLS (CONTINUED) 
 
 c ellipsoid semi-axis 
 
 f symbolic representation of source consisting of the volume dis- 
 
 tribution of induced magnetic dipoles 
 
 j FT 
 
 k relative permeability, ^- (in general, k = k' - j k" ) 
 
 k real part of permeability 
 
 k imaginary part of permeability 
 
 I length along the wire path 
 
 n number of turns on the winding 
 
 r distance from P to Q 
 
 s spacing between turns measured along the axis (^ a -) 
 
 u symbolic representation of source, consisting of assumed distribution 
 of induced magnetic dipoles in ferrite 
 
 w symbolic representation of the source consisting of the electric cur- 
 rents in the wire 
 z unit vector in z direction (direction of the a-axis) 
 
 3 free space propagation constant 
 
 |i permeability 
 
 M. Q |i for free space 
 
 A. free space wavelength 
 
 w circular frequency 
 
 : xy> reaction between the sources x and y. In general 
 
 -xy> - /// (E^JyiyCy) dV„ 
 
 : X y> - <yx" by the reciprocity theorem. 
 
1. INTRODUCTION 
 
 The recent development of ferrites having comparatively low loss and 
 high permeability has opened the possibility of improving the performance 
 of small loop antennas by the introduction of a ferrite core. In receiv- 
 ing antennas, which is the application we have in mind, the improvement, 
 if any, would appear as an increase in the signal-to-noise ratio for an 
 antenna of a given size. Since the signal- to-noise ratio is implicitly 
 determined by the input impedance, the essence of the problem is to cal- 
 culate the input impedance of a length of wire wound around a ferrite 
 core, for various core shapes and winding arrangements. 
 
 The typical antenna under discussion has a diameter of a few centi- 
 meters at 10 me. It follows that the contribution to the input resist- 
 ance due to ohmic losses in the winding can be treated separately from 
 the contributions due to core losses and radiation. The resistance due 
 to imperfect conductivity of the winding is therefore considered in a 
 separate study. 
 
 It is assumed that the ferrite is uniform, linear, and isotropic 
 For the sake of simplicity it is also assumed that the permittivity of 
 the core is that of free space, an assumption which, though incorrect, 
 has only a slight effect on the results. 
 
 It is apparent at the outset that it would be impractical to at- 
 tempt an exact solution based on Maxwell's equations- We therefore use 
 the reaction concept 1 to derive approximate formulas based on certain 
 assumed field distributions. The method to be discussed is based on an 
 assumed distribution of magnetic field in the core. Since the antenna 
 is electrically small, a good approximation is obtained by using the 
 magnetostatic solution as the assumed distribution,, This method is 
 therefore practically restricted to ellipsoidal shapes because it is 
 only for such shapes that the magnetostatic solution Cas represented by 
 the demagnetization factor") is available. 
 
 The formulas used have the usual stationary properties. At times 
 this fact is hardly relevant since the assumed distributions are some- 
 times nowhere near correct. What is interesting is that, despite a 
 gross error in the assumed distribution, the radiation' resistance is 
 given correctly by the formulas Indeed, it should be noted that the 
 radiation resistance R R( can be calculated with good accuracy by using 
 
 -I- 
 
a much simpler approach than that given in the following sections for the 
 calculation of the impedance (i.e., reactance and resistance due to radi- 
 ation and core losses). In outline, this simple method proceeds as fol- 
 lows. The antenna is represented as a magnetic dipole. The reciprocity 
 theorem is used to find the dipole moment in terms of the loop current 
 and the distribution of magnetic field, H , set up in the core by an in- 
 cident plane wave. The power radiated by the dipole (which can be easily 
 derived from a distant field calculation) is then set equal to the input 
 power %|l| Rp, thus giving a formula for Rp in terms of the distribution, 
 HL.. The result for a single turn loop is 
 
 Rr 
 
 31.171 
 
 #* 
 
 S 
 
 k H c 
 4 ~"° 
 
 dA 
 
 (1) 
 
 where 
 
 A. = free space wavelength 
 
 H Q = incident magnetic field (which is taken normal to plane of loop) 
 
 H c = component normal to plane of loop of magnetic field in core due 
 to incident plane wave 
 
 A = area of loop 
 
 k = relative permeability, lVn Q , of core (k in general is complex), 
 and the usual eJ wt time convention is implied. The quantity H C /H Q is 
 practically the same as for the magnetostatic case (because the antenna is 
 electrically small). When the shape of the core is ellipsoidal, the static 
 value of H c /H Q is a constant. Then Eq. 1 reduces to 
 
 2 
 _ (2) 
 
 Rr = hoii 
 
 whe] 
 
 1 + 19 (k - 1) 
 
 J9 = demagnetization factor (from static solution) 
 n - number of turns. 
 
 Thus, a ferrite core which completely fills the loop increases the 
 radiation resistance by a factor of |k/ [1 + JMk - 1)] | a « For many fer- 
 rites, this factor can easily be made of the order of a thousand. Even 
 so, for typical antenna sizes, the radiation resistance is only of the 
 order of 10 -6 ohms at 10 mc/sec 
 
 The formulas for the impedance are obtained from the basic relation 
 
 = 2* 
 
(which is derived in Appendix I) 
 
 I 2 Z = \ 
 
 wherein 
 
 fi 
 
 I = input current at the antenna terminals 
 
 Z = input impedance, 
 
 J is the surface density of electric current on the surface, S, of 
 the metal part of the antenna, and E is the electric field that would be 
 generated by ^J if said metal part were removed, leaving free space in its 
 place. (It is assumed that the "metal" is a perfect conductor and that 
 I, Z, J, and E are complex according to the ep wt time convention ) 
 Equation 3 is exact provided the terminals are an infinitesimal distance 
 apart and located at opposite sides of a gap in the metal structure. If 
 the antenna consists of a metal loop of wire, the surface, S, is the sur- 
 face of the wire, J is the surface density of current on the wire and thus, 
 ignoring the insignificant contribution from the gap, 
 
 S 
 
 J°E dS = #1 (Z) E t dZ 
 S 
 
 where I(Z) is the line density of current as a function of distance, Z, 
 along the wire and E t is the component of E in the direction of the dif- 
 ferential increment, dZ, (note that E is the field generated by I(Z) with 
 the wire removed) „ For the ^electrically small" case, I(Z) is practically 
 constant and equal to I , so that Eq„ 3 becomes 
 
 IZ - - f9E t dZ„ (4) 
 
 If the wire consists of a coil of several turns, Eq. 4 still applies, pro- 
 vided 5 is understood as following the wire from one input terminal to the 
 other. If the turns are closely spaced, E t is practically the same for 
 each turn and Eq. 2 reduces to 
 
 IZ = - n IOE t dZ (5) 
 
 where n is the number of turns. Equations 4 and 5 are well-known in con- 
 nection with conventional loop antennas. 
 
 -3- 
 
2 . FORMULAS BASED ON AN ASSUMED DISTRIBUTION IN THE CORE 
 
 The field generated by J in the presence of the core is the same 
 as the superposition of the fields generated in the absence of the 
 core by J and the volume density of magnetic dipoles (or currents), 
 jw(|i-|j, )H, where H is the magnetic field generated by J in the presence 
 of the core and \i and |i Q are the permeabilities of the core and free 
 space, respectively. To distinguish these two sources, let w denote 
 the wire electric currents J w = jJ (surface density) and f denote the 
 ferrite magnetic dipoles Kf j jd)(^-|i )H (volume density). Let E w , Ji w , 
 Ef, Hf denote the corresponding fields in free space, and E,H the 
 field of w in the presence of the core. 
 Then 
 
 E " l w + E f 
 
 and Eq. 3 can thus be written 
 
 )dS (6,) 
 
 I 2 2 - -J J J w • (E w + Ef : 
 
 s 
 or using the notation of the reaction concept 
 
 t2 7 ^ _ < ww > - < w f> (7) 
 
 x ** ww no core W1 no core v ' ' 
 
 and similarly Eq. 3 can be written 
 
 I 2 Z ■= - <ww> wlth core . (8) 
 
 On the assumption of uniform current in the winding, J_ is practically 
 the same with or without the core for a given input current. 
 
 ww no core l ^o Ky ' 
 
 where Z Q is the input impedance with no core. Therefore Eq. 7 takes 
 the form 
 
 I 2 (Z - Z Q ) - - <wf> no core . (10) 
 
 -4 
 
Assuming that Zr is known, the problem therefore reduces to finding an 
 approximation for <wf>. Thus, this approach consists essentially of 
 finding the extra effect of the core, assuming that the impedance with 
 no core is known. 
 
 To find an approximation from <wf> we start with an assumed volume 
 distribution, denoted by K a , which we are going to use as an approximation 
 to K£„ By using the exact relation K£ = jw(u.-|i „) (H w + Hf) (which follows 
 from the definitions of the f and w fields) a variety of different formulas 
 can be obtained for <W£> (see Appendix II). If, in these formulas, we re- 
 place the f_ field by its approximation, the a field, the different formulas 
 no longer give the same result- -they are all different approximations for 
 <W£>. The optimum approximation is therefore obtained if we can adjust 
 a to make all of these different approximations give the same result. 
 If we rule out those forms which require an explicit knowledge of H w , the 
 free space magnetic field of the loop--a complicated and practically 
 unmanageable function in most cases --we are left with three different forms 
 which can be made to give the same result by adjusting the level of the 
 assumed K a . The equation so obtained is (see Appendix II) 
 
 <wa p > = - <a p a p > 
 
 Ka • K £ 
 
 jw(|i-|i a )' 
 
 dV 
 
 (ii) 
 
 SW E * ds §5 * 
 
 K s 
 
 jw(li-M-o) 
 
 dV a 
 
 (12) 
 
 Making use of this result and replacing <wf> by <wa D > in Eq. 
 obtain the following approximation for the input impedance: 
 
 z-z 
 
 IIS, 
 
 ff'f v 
 
 J J J core — u 
 
 jd>(n-|j. ) 
 
 I u dV 
 
 10 
 
 (13) 
 
 The u field is the assumed distribution- -it differs from the a_ field 
 
 by a constant. (The distinction between the u and a fields is that 
 the level of the u field does not matter in Eq. 13 whereas the level 
 of the a field, if we use it in place of the £_ field, does matter. 
 (Equation 13 automatically ensures that a p takes on the level determined 
 
 -5- 
 
by Eq. 12.) For a loop consisting of a single turn 
 
 ff s J°£dS= - I # E t dZ (see Eq. 4') 
 
 = - I // A CV x E) N dA 
 
 where A is the area of the loop and N the direction normal to A so that 
 the current is right-handed about N 
 
 = I //a (J w '- L o + K)n ^ (Maxwell's equation). 
 Thus the problem of having to find E u can be avoided by writing 
 
 U/I) SI S J w E u dS 
 
 in the form 
 
 // A (jWMo H u + K U ) N dA. (13A) 
 
 For a distributed winding, the same device can be used to a good 
 approximation, by applying this relation to each turn, ignoring the fact 
 that the beginning and end of a turn are not quite at the same point. 
 Thus,, for a closely spaced winding, uniformly distributed along the 
 axis over the entire extent of the core, 
 
 U/I) // s J w . E u dS 
 can be replaced by 
 
 JL SH V (jupo H u + K U ) N dV (13B) 
 
 2a 
 
 where n is the number of turns and 2a the length of the core measured 
 along the axis of the winding. 
 
 -6- 
 
3. FORMULAS IN TERMS OF THE DEMAGNETIZATION FACTOR 
 
 We note that in order to use the formula (Eq„ 13) for Z, we must 
 find the magnetic field H u due to the assumed volume density of magnetic 
 dipoles, K u , [The electric field E_ u is not required provided the wind- 
 ing of the loop allows us to use the transformations (13A) or ( 13B) , ] 
 The aim of this section is to describe the calculation of H^ and its re- 
 lation to the demagnetization factor.. 
 
 The magnetic field, H£, generated in free space by a volume density, 
 K, of magnetic current (or dipoles) is given by 
 
 jwu H f (Q) - V Q x v Q x /// K(P) ^^ dV (14) 
 
 where 3 is the free space propagation constant, r is the distance from 
 P to Q, P is a point in the source, and Q is the point of observation. 
 It will be noted from Eq, 13 that we need the value of H^ only at points 
 within V, Thus the largest value of r involved is the maximum dimension 
 of the core, which is small compared with the wavelength 2^/3° Expansion 
 of the exponential therefore gives a rapidly convergent series: 
 
 e'^ T 1 - P 2 r . 3 3 r 2 
 
 = - -jP - -r- + J -r- '" (15) 
 
 r r 2 6 
 
 The first term gives the static solution (corresponding to |3='0). The 
 second term contributes nothing* Thus if K is real, the first contribu- 
 tion to the real part of Hf comes from the fourth terra,, which happens to 
 give a particularly simple integrand. The first order approximations to 
 the real and imaginary parts of H.£ can therefore be obtained fairly 
 easily from Eq, 14 if the static solution is known. Thus we try to find 
 some K for which the static solution is known and which is as close as 
 possible to the correct K„ Obviously the best choice would be the solu- 
 tion for the given antenna when energized with DC current but unfortunately 
 this is too complicated except for a few special cases. The only really 
 simple static solution is that for an ellipsoid in a uniform magnetic 
 field, which gives a uniform resultant field inside the ellipsoid. To 
 utilize this solution we therefore assume a uniform K and restrict the 
 core shape to an ellipsoid. Then the contribution to Eq, 14 from the 
 "static" term can be written in the form, for points in V, 
 
 Jum, o H.£ = - J9K (K is constant) (16) 
 
 .7= 
 
where $ is the demagnetization factor, the value of which is known from 
 the static solution: 2 it depends on the shape of the ellipsoid and is 
 independent of \i (see also Figs. 1 and 2). Thus, taking K to be real, 
 
 Im H f = J^ [1 + 0((3L) 2 ] (17) 
 
 o 
 
 and from the r term in Eq„ 15 one obtains 
 
 K6 3 V 
 Be H f - -"=—-[! + 0((3L) 2 3 (18) 
 
 where L is the largest dimension of the core, Substitution from Eqs 17 
 and 18 in Eq„ 13 gives 
 
 z - z ^ AV ° (k : l) (1 ~ J9) (j [1 : J9] - ^ [1 + r-^-A-^TT ^ ) < i9) 
 
 o V (1 -:- J9 [k - 1]) 
 
 for an ellipsoidal core with a single loop located in any plane of symmetry, 
 where 
 
 k =: iVMv. = relative permeability of the core 
 A = area of the loop 
 V = volume of the core 
 
 [It should be pointed out that if k is real the real part of Eq 19 
 which represents the increase in radiation resistance due to the ferrite, 
 agrees with the value obtained from the far field method of calculation 
 (see Eq. 1)] . 
 
 Similarly, for a winding of many turns uniformly spaced along an axis 
 of the ellipsoid 
 
 Z " Z ° ~ l2a) 2 U + J9(k"~ f)] (jI ] 6. U 1 + Hk - 1) J) 
 
 (20) 
 where n = number of turns 
 
 2a - length of said axis,, 
 
 This result serves as a check on the imaginary part of {Z - Z Q ) because 
 when n-*=° the assumed field becomes correct in the static approximation. 
 Therefore if k is real the imaginary part of (Z - Z Q ) should agree to 
 0((3 O 2 with the (exact) static calculation of inductance, which it does 
 
 -8 = 
 
Figure S Demagnetization Factors of Ellipsoids along the a Semi-Axis 
 
 -9- 
 
2 3 4 
 
 c/o Ratio of Semi-exes 
 
 Figure 2 Demagnetization Factors of Ellipsoids along the 
 a Semi -Ax is 
 
 10 
 
(see Eq. 22, Appendix IV, and also Reference 4,)„ 
 
 It should be noted also that k, in Eq. 19 or 20, can be either real 
 or complex.. Thus, since it has been found possible to represent the be- 
 havior of ferrites in terms of a complex permeability, 3 these equations 
 give the effect on input impedance of cere losses., 
 
 To find the actual value of the impedance from Eqs .. 19 or 20, the 
 impedance of the coil in free space, Z , must be known, The value to be 
 used in Eq 19 (the free space impedance of a single turn loop) is 
 
 2 
 
 Z oL " i w ^o b 
 
 In- 
 
 16b 
 
 2] 
 
 31,171 
 
 rcb* 
 
 X 2 
 
 (21) 
 
 in which 
 
 b = the radius of the loop 
 
 and 
 
 d ~ the diameter of the wire, 
 (The greatest accuracy is not to be expected for the reactive part of the 
 impedance found from Eqs, 19 and 21 since the assumed distribution is a 
 poor approximation to the actual,) The value of Z to be used in Eq„ 20 
 
 j(0U o n 2 (l - J9)V 31,171 
 (2a) 2 '* X 4 
 
 nV» 2 
 
 2a 
 
 (22) 
 
 in which 
 
 the semi-axis of the ellipsoid which is perpendicular to the planes 
 of the turns. 
 
 As was pointed out earlier, the quantity, J9, to be used in the formulas 
 can be found in the literature. It can also be found from the equation 
 
 J9j 
 
 l tl 2 cl 3 
 
 J (s * 
 
 ) 2 fr 
 
 ) ( 
 
 '&)'+' Ka + W 
 
 , J = 1, 2, 3 
 
 in which a- 
 
 l~, are the semi-major axes cf the ellipsoid and J9- is the 
 th 
 
 demagnetizati _>n factor along the j semi-axis. For convenience, graphs 
 which show the effect of the shape on the demagnetization factor are pre- 
 sented (Figs, 1 aid 2), The quantity J9 which is plotted in these figures 
 
 is really 
 
 th? 
 
 ilue along one of the axes- -namely, it is the value 
 
 -11= 
 
 MI1VERS1TY CF ILy&UJiJ 
 
along the axis which is perpendicular to the planes of the turns of the 
 winding. The curves are symmetric about the 45° line. 
 
 It is interesting to consider the ratio of the radiation resistance 
 to the resistance introduced by core losses for these ellipsoidal anten- 
 nas. It follows from Eqs. 20 and 22 that, if k = k' - jk" , 
 
 rad - 3 ■ X 6 
 
 R core loss k" (1 - J9) 2 
 
 assuming that k" << k (which is almost always the case). The variation 
 of this ratio with J9„ assuming constant value for the other parameters 
 (especially antenna volume) is shown in Fig, 3. In addition we note the 
 following: The ratio is: 
 
 1. directly proportional to the antenna volume, given a constant J9 
 and a particular ferrite. 
 
 2. directly porportional to the real part of the permeability, given 
 the antenna volume and shape and a specified ferrite loss tangent 
 
 3. directly proportional to the inverse of the loss tangent, given 
 permeability, shape, and volume. 
 
 12- 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 N 
 'O 
 
 X 
 
 lO o 
 K Q _ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 sso-i aaooj 
 
 -13- 
 
REFERENCES 
 
 1. Rumsey, V.H. Phys. Rev. 94, 1483, 15 June 1954 
 2„ Osborn, J„A„ Phys. Rev. 67. 357, 1945 
 
 3. Salpeter, J.L Proc. I.R.E. 42, 521, March, 1954 
 
 4, Weeks, W„L„ Technical Report No. 6, Antenna Section, Electrical 
 Engineering Research Laboratory, University of Illinois, Urbana, 
 Illinois, 20 August 1955. 
 
 14 
 
APPENDIX I 
 
 DERIVATION OF THE FORMULA I 2 Z ■ - // JE dS 
 
 s ~ ~ 
 
 The formula I 2 Z ■ - // J°E dS has been used for decades (in special 
 forms) but there is considerable confusion about it in the literature, 
 
 particularly in connection with an alternative formula which uses the 
 
 2 * 
 
 idea of complex power, i.e.,, jl| Z = // J°E dS where the star denotes 
 
 S 
 complex conjugate, Since the formulas have always been used for approxi- 
 mate calculations based on an assumed real J, the difference, if any, 
 between the two formulas has? never been uncovered. Since the first 
 formula is valid in general for any perfectly conducting structure, as 
 we shall shortly prove, in general J is not real and the two formulas are 
 therefore distinctly different/ There is a possibility that they might 
 give the same Z but this appears to be very difficult to decide. Since 
 we can demonstrate the correctness of the first formula, but not of the 
 second, it is the first which we use. 
 
 The essence of the problem is to calculate the circuit theory param- 
 eter Z by using field theory. The key to the problem is therefore the 
 connection between circuit theory ar.d field theory. To establish the 
 appropriate connection suppose that the antenna is energized by a constant 
 voltage generator. Then we have to find the field theory source which is 
 equivalent to a constant voltage generator* To do this we note that since 
 the problem implies the existence of the circuit parameter Z, the terminals 
 of the antenna must take the form of two points, A and B, an infinitesimal 
 distance, l ? apart, at opposite sides of a gap in the metal structure of the 
 antenna, as illustrated in Fig. 4. Since it will turn out that we have to 
 assume perfectly conducting metal in order to justify the formula, let us 
 make that assumption now in order to define the problem more clearly. Now 
 consider a uniform surface density JK volts per meter of magnetic current 
 flowing around a perfectly conducting cylinder, henceforth called the "plug", 
 which is placed between the terminals as in Fig. 4. Since tangential E 
 vanishes at the surface of the plug and is discontinuous at the magnetic 
 current sheet by the amount K, it follows that 
 
 E - n x K 
 just outside of the magnetic current sheet, n being a unit vector in the 
 direction of the outward normal, as shown in Fig. 4. Since I is infin- 
 itesimal we can relate E. to the circuit concept of input voltage, I), by 
 
 = 15- 
 
Antenna 
 ( Conducting 
 Metal ) 
 
 Conducting Plug 
 
 Figure 1 Antenna Energized by Magnetic Current Source (Equivalent to 
 a Constant Voltage Generator) 
 
 the equation 
 
 A 
 
 I) * U A '-U B » - / E : -ai =- -El = - Kl, 
 
 the directions of E and K being as shown in Figo 4„ Thus if K_ is fixed, 
 U is fixed: the field theory equivalent of a constant voltage generator 
 is a uniform solenoid of magnetic current of surface density 0/ I , com- 
 pletely filled with a plug of perfect electric conductor, of length I 
 and diameter d, where d/Z-o and l-o. 
 
 Note that the addition of the plug is not trivial for, according to 
 circuit theory, when a constant voltage generator is switched off, the 
 internal impedance between the terminals is a short circuit, and according 
 to field theory, when jC is switched off, the terminals remain connected 
 by the plug, which is a short circuit,, Plainly^ the internal impedance 
 would not be a short circuit if the plug were absent„ 
 
 The problem of a constant voltage generator connected to the antenna 
 can thus be defined in terms of field theory as the! problem of the source 
 represented by K in the presence of the short-circuited antenna. Let S 
 denote the entire surface of the perfect conductor which is formed by the 
 
shorted antenna, i„e„, S is the combination of the metal surface of the 
 antenna and the surface of the plug: (S is a single closed surface if 
 the antenna consists of two metal parts between which the terminals are 
 situated). Let h denote the distribution of magnetic current, K, and c 
 denote the electric current distribution which is induced on S by h„ 
 Thus c denotes the combination of the antenna current distribution and 
 the distribution of the current on the plug; it consists of a surface 
 density ^J which is continuous at every point on S„ (Note that the an- 
 tenna current is not continuous everywhere because it stops at the in- 
 put terminals) . Now it follows from Maxwell's equations that the field 
 due to h in the presence of the shorted antenna is identical to the super- 
 position of the fields due to h and c in the absence of the shorted anten- 
 na, Symbolically, this can be expressed in the form 
 
 Ihl = EhO * E c0 (i) 
 
 ftil = MhO- + HcO (ii) 
 
 where subscripts c and h denote the source of the field in question and 
 the subscripts and 1 represent the environment which exists when the 
 antenna conductor is removed and when it is in place, respectively. (If 
 the antenna consists entirely of the metal parts involved in the definition 
 of S, then represents free space). 
 
 We now bring in the reciprocity theorem by multiplying {ii) throughout 
 by _K, the surface density of magnetic current which comprises the source h, 
 Note that 
 
 M H hl • K dS - I! (H h0 • K * H c0 . • K) dS (iii) 
 
 over K_ over K 
 
 or, to state this result more precisely, 
 
 // Shi • dK = // H h0 ■ dK + // H c0 ■ dK. (iv) 
 
 Now since d/i-0 and-i-G, H hl and H^q are parallel to K at points occupied by 
 K and since K is uniform (K = -U/0, 
 
 // Hh • dK -U £ H h 
 
 dS 
 
 where ^>dS represents integration around a typical loop of the magnetic cur- 
 rent solenoido Thus, from Amperes law 
 
 // fl bl - dK- -1)1 
 -17- 
 
where I ~ input current as in Fig„ 4, and 
 
 // H h0 ' dK - 
 
 since H^q tends to the static theory limit of zero, as I - and d/l - 0, 
 Also from the reciprocity theorem 
 
 If HcO • dK - - ff E hQ ■ dJ 
 
 where _J is the surface density of electric current which comprises the 
 source c„ Substitution of these results in (iii) gives 
 
 1)1 = ff_^ E h0 ■ dJ, 
 
 Finally we bring in the boundary condition that the tangential com- 
 ponent of E^ vanishes at S„ Therefore, from (i ) , 
 
 N < E h0 r - N x E c0 on S 
 
 where N is a unit vector pointing in the direction of the outward normal 
 to S„ Since (v) involves only the tangential component of E^q we have, 
 therefore, 
 
 01 =■-// E c0 ■ dJ 
 
 or I 2 Z - 'U 2 Y = -// E J dS 
 
 s 
 
 where E_ is the electric field due to J^ in the absence of the perfectly 
 
 conducting parts of the antenna, and J_ is the current distribution on 
 
 the conductor formed when these parts are shorted at the input terminals 
 and excited as illustrated in Fig, 4. 
 
 When applied to the ferrite loop antenna we take S as the surface of 
 the shorted loop„ \he presence of the ferrite core makes no difference to 
 the argument. 
 
 The part of the analysis which begins with Eqs (i) and (ii) can 
 alternatively be expressed more elegantly in terms of the reaction concept 
 as follows. From (ii) we have 
 
 \)1 = < hh >{ ■= < hh >q : ' < he '0 
 
 < hh > - as l - °» 
 < he >« = - < cc > q from the boundary condition at a 
 perfect conductor. , l2z = U 2 Y . W - - < cc >„. 
 
 -18- 
 
APPENDIX II 
 DERIVATION OF EQUATION 11 
 
 As was stated earlier, one can write several different formulas for 
 the reaction, < wf >. Different approximations for < wf > are obtained 
 by replacing 1 by a reasonable assumption. The optimum approximation is 
 obtained by adjusting the level of the assumed distribution, a to make 
 the different approximations give the same result. This procedure leads 
 to Eq. 11 in the text. Explicitly, the reaction between w, the electric 
 currents in the windings, and f, the magnetic dipoles in the ferrite, 
 can be expressed in any of the following four forms: 
 
 $ J Jav ° J?f dS by definition /-\ 
 
 o ye r w 
 
 -HI Kf ° I^dV by definition (^ 
 
 er f 
 
 'III ju(n-H Q ) (H f + HJ • Il^dV which follows from (iii) 
 
 over core 
 
 K f - ju(n-u ) (H„ + Hf) 
 
 over core |J O 
 
 -/// K f • t^p— r a f 
 
 dV which follows from the same relation. 
 
 (iv) 
 Note that each form is expressed in terms of two vectors:- (i) J w , Ef; 
 
 (ii) _K f , H w ; (iii) Hf, f^; (iv) Kf, Hf. All of these forms are equal 
 as they stand but if we replace the correct source, f, by its approximation, 
 a^, they are no longer all the same. The reciprocity theorem ensures chat 
 (i) and (ii) are still the same, but that is all. We thus obtain three 
 different approximations for the reaction < wf > by replacing f with a p . 
 The optimum approximation would thus be obtained if we could so choose a p 
 to make all three the same. Considering that we have only one degree of 
 adjustment at our disposal (the level of a_), the best we can do is to 
 make two of them the same. The decision as to which two should be made 
 the same can be settled by inspection of (i), (ii), (iii), (iv). Note 
 that J w is known indeed we assume it to be a constant line density I 
 
 -19- 
 
but H^j, the magnetic field of the loop in free space, is not known ex- 
 plicitly. If we start from an assumed K.£ then in principle Hf and Er 
 are not known explicitly either, but if we take Kr as the magnetostatic 
 distribution corresponding to the problem of the core in a uniform field, 
 and if the core is ellipsoidal, then H£ and Er can be calculated very 
 much more simply than F^. Thus, as a practical matter, we are forced to 
 reject (ii) and (iii) and hence the optimum approximation is given by 
 equating (i) and (iv) which also agrees with (ii) because of the re- 
 ciprocity theorem. Replacing f by a in (i) and (iv) then gives Eqs. 11 
 and 12. 
 
 A further point in favor of (iv) rather than (iii) is that (iv) gives 
 a stationary formula for the impedance whereas (iii) does not. This is 
 demonstrated in Appendix III. 
 
 -20- 
 
APPENDIX III 
 STATIONARY PROPERTIES 
 
 To demonstrate the stationary properties of Eq„ 11, let6a represent 
 iation of the approximate source, a „ Sine 
 pedance is given by Eq„ 10 with a in place of f: 
 
 a variation of the approximate source, a „ Since the approximate i 
 
 I 2 (Z -Z ) ■- < w 3 a > (10) 
 
 We have 
 
 I 2 6 Z - < w, 6a > 
 
 where 6 Z represents the variation in Z due to the variation in a„ Now 
 < w, 6 a > is given by Eq„ 11 in the form 
 
 <w, 6 a > = — < a, 6a > -<6a a > — < 6a 6a > 
 rrr 26 K a ■ K a w rrr 6K a • 6K a dV 
 
 - /// jurrt-nf dV - M -jt-nr-^-fr-r 
 
 core 
 
 (which follows directly from the fact that both a and a + §a must satisfy 
 Eq„ 11, ) By the reciprocity theorem 
 
 < a, 6a_ > = < 6a a > 
 
 Also, for a variation about the correct solution, we can substitute the 
 correct source f for a in the formula for the variation. Thus we can 
 replace 
 
 by 
 
 jw ( \x - n Q ) 
 
 K f---V ■ I!f + Bw 
 
 j(j(|i - |i Q ) 
 
 iij -rwfV^S-r := SSI 26 K a ■ (H f + H w ) dV - -2<6 a, f> -2<6 a, w>. 
 
 core J ^ O 
 
 o 
 
 26 K Q • K Q dV 
 
 -21- 
 
Substitution of these relations in the formula for < w, 6 a > gives 
 
 < w s 6 a > = -2< 6 a, f > - < 6 a, 6 a> + 2< 6 &, f> + 2< 6 a, w > 
 
 < w, 6 a > = < 6 a, 6 a>„ 
 
 Thus the variation of Z is of the second order for variations of a 
 about the correct distribution. 
 
 On the other hand, if we used (i) and (iii) of Appendix II to 
 obtain an approximate formula for Z, the formula for a would take 
 the form (on replacing f by a) 
 
 < w a > = - /// jw(u - u Q ) (H a + %) • \ dV. 
 
 core 
 Now 
 
 < w 6a> = - /// ju(u - u ) 6 H _■■ H dV 
 
 core ° a "W 
 
 which is of the first order, since 6 H a is an arbitrary variation, 
 independent of H^„ The formula so obtained therefore is not stationary, 
 
 -22- 
 
APPENDIX IV 
 
 CALCULATION OF INDUCTANCE USING STATIC THEORY 
 
 According to static theory, a winding of n turns on the surface of 
 an ellipsoid uniformly spaced along an axis gives a uniform internal 
 field as n-°° In the limit the winding can be represented by a 
 surface density. X It is found that _J is related to the internal 
 field H, by the equation 
 
 j = H x N l±Mkzll_ d) 
 
 - - - 1-J9 
 
 where 
 
 19 = demagnetization factor 
 
 N = unit vector in direction of outward normal. 
 
 To find the inductance, we can start with the formula for the 
 input voltage, l); 
 
 D = - #E • dl 
 the integral being taken along the wire and across the gap. Thus 
 
 U - 2 -f. K - dZ 
 
 all turns round each turn 
 
 2 -// V x E ■ £ dS as n-c° 
 
 all turns over each cross section 
 
 where _z represents a unit vector in the direction of the axis 
 
 = 2 //juii H z dS 
 
 oyer each cross section 
 
 - 1 Iff jw^H - zdV 
 
 volume of core 
 
 - L //,/ jw(i H dV since H is parallel to z„ (ii) 
 
 ■volume of core 
 where 
 
 s - spacing between turns, measured along the axis 
 
 -23- 
 
where 
 
 2a = length of axis. 
 
 Now. to express the input current, I, in terms of H, note that by 
 definition 
 
 s J =?xNl, 
 Comparison with Eq» (i) shows that 
 
 I . sH [l-^(k-l U. (iii) 
 
 1 - 19 
 
 Since H is constant (ii) gives 
 
 U = 1 jup, HV (iv) 
 
 s 
 
 where V = volume of core. 
 
 i = JL. = u V 1 - J (v) 
 
 Jul s i (l+J9[k-l]) 
 
 It can easily be verified that this is consistent witn Eq. 20. 
 
 -24- 
 
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