United States Patent
`Herman
`
`[19]
`
`3,956,751
`(11)
`[45] May 11, 1976
`
`[54]
`
`MINIATURIZED TUNABLE ANTENNA FOR
`GENERAL ELECTROMAGNETIC
`RADIATION AND SENSING WITH
`PARTICULAR APPLICATION TO TV AND
`FM
`
`[76]
`
`Inventor:
`
`Julius Herman, 3906 Bel Pre Road,
`Silver Spring, Md. 20906
`
`[22]
`
`[21]
`
`[52]
`
`[51]
`[58]
`
`[56]
`
`Filed:
`
`Dec. 24, 1974
`
`Appl. No.: 536,168
`
`US. Chie ceeeeeeeeeeees 343/744; 343/882;
`:
`343/895
`Int. Cn? neces seceeceeeeneeeeteees HO01Q 1/36
`Field of Search........... 343/742, 743, 744, 895,
`343/748, 882
`
`References Cited
`UNITED STATES PATENTS
`
`3,005,982
`3,261,019
`3,534,372
`3,560,983
`3,780,373
`
`Bernfeld ...........ceeeeeeeeeee 343/742
`10/1961
`T4966 Lundy... eeeeeereeneee 343/742
`10/1970
`Scheuereckeretal............. 343/742
`2/1971 Willie et al. 343/744
`
`12/1973 Holst... ceeeeeseeesesceeeeseeeeeee 343/895
`
`Primary Examiner—Eli Lieberman
`Attorney, Agent, or Firm—Cameron, Kerkam, Sutton,
`Stowell & Stowell
`
`ABSTRACT
`(57]
`A miniaturized thin-wire multi-turn series-connected
`helical loop antenna, the volutes of which are concen-
`tric turns, closely spaced so as to exhibit strong mutual
`coupling effects. A lumped impedance, fixed or vari-
`able, is disposed electrically in series with one of the
`turns, and a selectively actuable multi-position switch
`interconnects the turns and the lumped impedance to
`maximize efficiency.
`
`2,657,312
`
`10/1953
`
`Saranga ......... eee eeeeeeeee 343/744
`
`19 Claims, 8 Drawing Figures
`
`#24 Enameled
`Magnet Wire
`
`
`
`Infineon Exhibit 1030
`Infineon Exhibit 1030
`Infineon v. AmaTech
`Infineon v. AmaTech
`
`

`

`
`
`U.S. Patent May 11,1976=Sheet 1of 4 3,956,751
`
`
`=180°
`
`Fig. 2
`
`

`

`U.S. Patent May 11,1976
`
` Sheet2o0f4
`
`3,956,751
`
`g=180°
`
`
`

`

`U.S. Patent May 11,1976=Sheet 3 of 4.==.3,956,751
`
`37
`Dp
`
`Oo
`
`fp
`
`3
`
`WT
`
`NIKE Nag
`ty >
`2) 3
`as ig
`an
`|
`
`2
`
`vl
`
`.
`Fig. 5
`
`x
`
`

`

`U.S. Patent May 11,1976
`
` Sheet4of4
`
`3,956,751
`
`#24 Enameled
`
`Magnet Wire
`
`

`

`1
`
`3,956,751
`
`MINIATURIZED TUNABLE ANTENNA FOR
`GENERAL ELECTROMAGNETIC RADIATION
`AND SENSING WITH PARTICULAR APPLICATION
`TO TV AND FM
`
`BACKGROUNDOF THE INVENTION
`1. Field of the Invention
`The subject invention relates to miniaturized multi-
`turn series-connected loop antennas.
`2. Description of the Prior Art
`The theoretical solution for multi-turn loop antennas,
`whethertransmitting or receiving, is initially based on
`the solution for a single-turn loop. The single-turn solu-
`tion can be obtained by considering the antenna to be
`a transducer which converts concentrated or distrib-
`uted voltages into distributed fields and vice-versa. The
`phenomenon which accomplishes this is the flow of
`current on the antenna conductor. Thus, if one can
`postulate the true form of the current andtheresulting
`fields in space in response to timevarying driving
`forces, the solution for the single-turn loop, whether
`transmitting or receiving, can be obtained.
`A previous patent issued to me on Feb. 19, 1963,
`U.S. Pat. No. 3,078,462, was titled ‘““One-Turn Loop
`Antenna”. This patent discloses the basic solution for a
`thinwire, one-turn loop antennain air, without and with
`inserted impedances. Although the one turn loop an-
`tenna described in the aforesaid patent was a distinct
`advanceoveravailable loop antennas,its limitation toa
`single loop design subjects it to several limitations. The
`extension of the theoretical analysis of the operational
`characteristics to multi-turn loop antennas had not
`been accomplished, probably because of the difficulty
`in developing a solution which takes into account the
`interconnection discontinuity between turns of a multi-
`turn series connected loop antenna having concentric
`planar turns.
`Without such a usable theoretical analysis, prediction
`of the operation of a multi-turn loop, or the effects of
`various modifications thereof are well-nigh impossible.
`As a result, the development of loop antennas today
`has not progressed much beyondthe level of my afore-
`said patent.
`I have now developed a complete mathematical the-
`ory for the characteristics of a multi-turn, series-con-
`nected loop antenna formed of planar turns that are
`closely spaced and exhibit strong mutual coupling ef-
`fects. The theory from which the multi-turn loop has
`been developed is applicable not only to miniaturized
`antennas, but to any electrical size. Hence, the ability
`to control radiation coverage and antenna impedance
`of a transmitting antenna, and the ability to control
`responseto incident fields and the antenna impedance
`of a receiving antenna, apply also to an antenna of any
`electrical size constructed in accordance with the prin-
`ciples and teachings of the present invention.
`SUMMARY OF THE INVENTION
`
`The present invention relates to a multi-turn thin-
`wire. series-connected loop antenna formed of, planar
`concentric turns, cylindrical turns, or any three-dimen-
`sional geometric configuration, having turns that are
`‘closely spaced so as to exhibit strong mutual coupling
`effects. A lumped impedance,fixed or variable,is dis-
`posed at a selected point in the periphery of one or
`more turns, and a multi-position switch is intercon-
`nected between the turns and the lumped impedance to
`
`maximize efficiency. In accordance with the present
`invention, there is provided a miniaturized antenna
`capable of achieving high radiation efficiency such as
`50% or more for a miniaturized antenna whose maxi-
`mum dimension is less than 0.07 over a 10 to 1 fre-
`quency range and having an impedance that can be
`easily adjusted to almost any desired value. The radia-
`tion pattern can be selected to conform to many de-
`sired coverages.
`Whenutilized for reception, the miniaturized receiv-
`ing antenna converts incident fields into a maximal
`signal voltage at the input of a receiver over a fre-
`quency range of as much as 16 to 1 by utilizing the
`ability to control the antenna impedance.
`Being miniaturized, the receiving antenna can act as
`a probe in spaceso asto finely select and respond to a
`particular incidentfield in the presence of a numberof
`simultaneousincident fields. In essence, it can reduce
`the undesirable “ghost” effect. Such a receiving an-
`tenna respondsto both horizontally and vertically po-
`larized fields. Hence, its physical attitude can be easily
`adjusted to provide polarization sensitivity.
`Accordingly, it is an object of my invention to pro-
`vide a miniaturized transmitting antenna in a concen-
`tric multi-turn series-connected loop configuration,
`planar and non-pianar, smaller than any other transmit-
`ting antenna presently known, where the radiation
`efficiency is an appreciable fraction of 100%.
`Another object of my invention is to provide a minia-
`turized transmitting antenna in a multi-turn series-con-
`nected loop configuration capable of covering fre-
`quencybandsof as much as16 to 1: (a) by the use of
`a single tuning condenser inserted in a single turn of the
`multi-turn loop; (b) by the use of switches to switch
`opensto shorts and vice-versa in one or more turns.
`A further object of my inventionis to provide a trans-
`mitting antenna in which, regardless of size, a pre-
`scribed and desired field pattern and a desired imped-
`ance level can be easily adjusted by using inserted im-
`pedances, fixed and/or variable, in one or more turns of
`a multi-turn loop to select and/or repress particular
`current modes.’
`’ Still another object of my invention is to provide a
`multi-turn series-connected loop antenna configuration
`for
`radiation and/or
`reception of electromagnetic
`wavesin which turns can be connectedin single spiral
`form, in double spiral form, in reversed turn form, and
`in partial turn form, so as to achieve characteristics
`such as broad-banding, field pattern control, imped-
`ance level control.
`Another object of my invention is to provide a minia-
`turized passive probe antenna in a concentric multi-
`turn series-connected loop configuration, planar and
`non-planar, for reception of VHF black and white and
`color TV and FM, with maximum loop antenna diame-
`ter being less than fiye inches.
`A further object of my invention is to provide a min-
`iaturized passive probe antenna for VHF black and
`white and color TV and FM, capable of minimizing
`“ghosts” by universal adjustment of the plane of the
`multi-turn series-connected loop.
`Another object of the invention is to provide a minia-
`turized passive probe antenna for VHF black and white
`and color TV and FM which uses internal antenna
`tuning in the form of a single miniature tuning con-
`denser inserted in one turn of a multi-turn series-con-
`nected loop to maximize the signal voltage developed
`across the input impedance of the TV or FM receiving
`
`10
`
`20
`
`25
`
`30
`
`35
`
`40
`
`45
`
`50
`
`55
`
`60
`
`65
`
`

`

`3,956,751
`
`3
`
`set.
`A further object of my invention is to provide a min-
`iaturized passive probe receiving antenna capable of
`covering frequency bandsof as much as 16 to | by the
`use of switching configurations which change opens to
`shorts and vice-versa in one or more turns of a multi-
`turn series-connected loop antenna.
`A distinct and important advantage of the present
`invention is that the foregoing objects can be achieved
`through a frequency spectrum from a few hertz up to
`the point where physical limitations preclude practical
`embodiment which is about 5000 MHZ.
`
`BRIEF DESCRIPTION OF THE DRAWINGS
`
`These and other objects of the present invention and
`the attendant advantages will be more apparent and
`more readily understood upon reference to the follow-
`ing specification, claims and drawings wherein:
`FIG. 1 is a pictorial representation of the spherical
`geometry employed in formulating the theory of a one-
`turn, thin-wire loop antenna;
`FIG. 2 is a diagrammatical representation of a single
`turn loop antenna having an inserted impedanceZ, of
`the same value as the load impedance and being cross-
`connected;
`FIG. 3 is a diagrammatical representation of a single
`turn loop antenna having an inserted impedance Z, of
`the samevalue as the load impedance and parallel-con-
`nected;
`FIG. 4 is a pictorial representation of the geometry of
`a multi-turn, series-connected loop antenna;
`FIG. 5 is a schematic equivalent circuit for the multi-
`turn loop of FIG. 4 in the X—Y plane;
`FIG. 6 is a perspective view of a multi-turn, series-
`connected loop antenna embodying the present inven-
`tion;
`FIG. 7 is an enlarged fragmentary front elevational
`view of a switch interconnecting the several loops of a
`multi-turn loop antenna embodying the present inven-
`tion; and
`FIG. 8 isa perspective view of a universal joint suit-
`able for use with the present invention.
`DESCRIPTION OF THE PREFERRED
`EMBODIMENTS
`
`20
`
`25
`
`30
`
`35
`
`40
`
`45
`
`Inasmuch as an appreciation of the advance and
`contribution to the art madeby the present invention is
`dependent on an understanding of the mathematics
`behind the design, the basic solution of one-turn loop
`antennas and its modification and application to multi- 59
`turn series-connected loop antennaswill be discussed
`first.
`
`4
`
`BASIC SOLUTION OF ONE-TURN LOOP
`ANTENNA
`
`(i)
`
`Nb) =at
`
`(a, cos k@+b, sin kp]
`
`The spherical geometry shown in FIG. 1 is used in
`formulating the theory. The loop lies in the X—Y
`plane, the radius to the extreme edge of the loopis c,
`the wire radius is a, and the radius to the center of the
`loop conductoris 6. A driving voltage at @ = 0° causes
`a counterclockwise current flow around the loop. In
`general, the current flow can be caused by the single
`generator V,, by several of these generators atdifferent
`points on the loop, by a distributed generator such as
`whenthe loop is the secondary of a transformer, and by
`electric fields in space which are incident on the loop.
`The incorporation of any orall of these into the equa-
`tion neededto solve the problem will be shown subse-
`quently. In any event, the flow of current, I(@), is con-
`tinuous and hence maybepostulated as a Fourierseries
`in the variable ¢ as follows:
`oa
`2ke
`wherethe current coefficients for each mode, do, az, Dy
`are unknownasyet. For a thin wire, the current can be
`assumed as filamentary. The solution for the fields in
`the spherical continuum outside of the loop is now
`obtained using Maxwell’s equations which: relate the
`electric and magneticfield vectors to the scalar electric
`wave potential and the vector magnetic wave potential;
`relate the dependence between vector magnetic wave
`potential and scaler electric wave potential; and relate
`the dependence of the magnetic vector wave potential
`on the integral summation of current moments as modi-
`fied by the retardation Green’s function. The driving
`forces for the current are assumed to be time-depend-
`ent in the well known form of e”.
`Applying the addition and expansion formulas in
`spherical coordinates to the Green’s function, integrat-
`ing, collecting terms, substituting the magnetic poten-
`tial components in the electric and magnetic field equa-
`tions, and using the general spherical wave equations,
`the resulting fields are expressed as two families of
`waves, one a family of transverse magnetic waves, TE,
`and one a family of transverse magnetic waves, TM. In
`my aforesaid patent, the TM family was expressed as
`two families, a TM? and a TM®.I have found, however,
`that these two families can be consolidated into one
`TM family.
`,
`Theyare as follows:
`
`TE
`
`n
`a
`d
`.
`(n-k)
`(Bor)
`=
`(2n4+1) TO (Bob) I paK(cos@) Fa7
`k=0.1.2.... .
`Cateye
`MB
`Bor
`?
`qa
`wo
`n
`A(2 nt+t)
`Qo
`{n-k)!
`-
`n*(cos@)
`d
`.
`
`
`
`
`rEg — “a” n(rtl)rbk)! 1#(BebHa(Bor) aceraneade [Pn(cos8)]——[axsinkb—by coskd]2» :
`n=kt+1, K=1,2,...
`A+3,...
`6=90°
`n
`.
`2
`j
`-,
`(nek),
`kQQntl)
`5 2
`& nO) atk) Jnl Bob)An (Bor)
`n=kt+1, k=1,2....
`A+3,
`
`A=
`
`"
`
`.
`i
`2
`
`oa
`=
`nmktl
`A+3,
`
`rg =-—
`
`n*(cosé )
`[pn*(cos6)]
`
`.
`sink
`kot,
`[a,cosk@t+b, sink}
`6=90°
`
`k
`pr*(cos8)
`sind 7 [pn*(cosé)]
`
`;
`[a,sink@—b,coskd]
`6=90°
`
`rEg =—
`
`d
`d
`“
` (n-k)t
`(2nt1)
`t n
`2
`x
`= — Unrk)! JIn(Bob)Hal Bor) 7 pa*(cos8) 76 [px*(cos8)]
`_
`(nt+1)
`k=0, 1,2...
`n=k+l,
`.
`
`[a,cosko+bsinkd ]
`6=90°
`
`
`
`

`

`5
`|
`6
`rg = a Spector UO! Iy(Bob)Hn!(Bor) “>[pa*(cos8)] —— [pak(cos8)]
`
`:
`
`co
`
`n
`
`(Qnt1)
`, AFU,1,2Z,...
`n=
`KB
`
`3,956,751
`
`(nk)!
`
`.
`
`—_[a,cosk6-+b,sinks]
`
`6=90°
`
`™T™
`
`n
`
`x,
`
`+2.
`
`k(2n+1) or Jn"(Beb) aber,t(c058)pa8(o)Laysinkd—byc03k¢]
`
`n
`ot
`Hy (Bor)
`°
`
`a
`
`rE,=— “e z
`
`°
`n=k,
`k+2,
`
`te no
`k(Qnt1)
`Grek)!
`rEg = 77
`OS
`| , hGH) “Gaon ds"(Bob)Hn“BNg
`n=k,
`yaegeee
`k+2
`
`[Pa®(cos@)]pa'(o)Lasinkd—b,coskd|
`
`
`_(n-k)!
`
`(nth)!
`
`Jn(Bob)Hn(Bor) “yp[Pak(cosé)]pa*(o)[axsinkd—b,coskd]
`
`Qnt1)
`
` (n-k)
`
`“
`
`.
`(cos)
`
`/
`
`a
`
`n
`
`j
`
`k(2n+1)
`
`rHg = — x , t, honkl)
`
`n=k, A=1,2,...
`A+2,..
`oO
`
`n
`
`
`
`rEg = eng mart) “Gere Ia"(BobFie!Bor) Fapalo)axcosk-+bysinkg}
`ka. °
`saree
`sind
`
`-
`nN Qnth)
` (n-kyts
`-
`w(Cos8)
`.
`rig — “2 EE ety “GedyBab) BalBor) sing—PatCoVLaxcoskétbesink
`ktBe
`
`The dependence of the fields on the three spherical
`coordinates, [r,0,@], is that of a product of functions
`of each coordinate. Theelectric fields are represented
`by E, the magnetic fields by H. The nomenclature used
`aboveas follows:
`
`length and current, then the equality can be expressed
`as follows:
`
`or
`
`E@ et= Ed int = Z'1(@)
`
`[Ed ext —ZiID] =
`
`Ed totat — 0
`
`(2)
`
`No = impedanceof free space = 12,7onms
`j=
`—
`This equation is the same as that for a perfect conduc-
`Bo = propagation constant in free space = 27/A,
`tor in which the total tangential electric field at the
`= free space wavelength corresponding to the
`surface is zero. In essence, we have accounted for the
`frequency used
`~
`finite conductivity by replacing its effect by an electric
`Jn(Bob) = spherical Bessel function with argument (8,b)
`J,n'(Bob) = derivative of 5, (B8.b) with respect to the
`field which is a contribution to the total electric field.
`a
`argument (8b)
`H,,(B.t) = spherical Hankel function of the second kind
`This equation is a point relationship.
`with argument fo
`If an open,orafinite physically small impedance, or
`It is equal to [f,(Bor)-iNu(Bor)]
`a physically small driving voltage source, is placed in
`the loop periphery at a point, then at that point, there
`is a contribution to the total external tangential electric
`field in addition to that caused by the currentflow. This
`contribution can be accounted for in Eq.(2) by the use
`of the wellknown mathematical concept ofa slice or
`delta function generator or sink so that the electric
`field at the point exists only at the point or gap. This
`delta function generatororsink is defined asan infinite
`electric field in a region ofinfinitesimal length having a
`line integral across the region whichis finite and equal
`to the voltage at the point or gap. At the outer edge of
`the loop conductor where r=c and d=du, the delta
`function relation is:
`
`40
`
`45
`
`50
`
`55
`
`60
`
`65
`
`Hn(8,r) = derivative of H,(8,r) with respect to (8,r)
`P,*(cos@)= associated Legendre polynomial with
`argument(cos)
`The vector components of the electric and magnetic
`fields are shown in FIG. 1. The unknown constants in
`the field equations are the current coefficients of each
`current mode, a;, and b,. To determine them, it is nec-
`essary to resort to the boundary condition at a physical
`discontinuity in space, in this case, the extreme outer
`edge of the loop conductor (r=c, 0=90°). Assuming a
`perfect conductor, theelectric field, Eg , tangential to
`the loop peripheral surface must be zero everywhere at
`‘the surface. If this were not so, an infinite current
`would flow in response to a net electric field. If the
`conductor is not perfect, the tangential electric field
`“must be continuousacross the surface into the conduc-
`tor. This implies that the external electric field at the
`surface must be equal to the internal electric field at
`the surface. If the internal field is expressed equiva-
`lently as the product of internal impedance per unit
`
`Epo= Lc Vq?8(P—-,)
`
`(3)
`
`Taking the line integral across an infinitesimal gap
`extending from —6/2 to +8/2:
`,
`8/2
`8/2
`[
`Eg gap cd = I
`Vea? 8(6-6,)db = Vs
`
`8/2
`
`—8/2
`
`(4)
`
`

`

`3,956,
`
`751
`
`7
`The integral of a delta function limits the value of the
`integral
`to that of the function modifying the delta
`function,
`in this case V,°*". The integral is also zero
`everywhere but at the point d= ¢,. Obviously, if the
`modifying function exists only at ¢= 0°, the delta func-
`tion would be 6(¢@). The delta function, 8(¢—¢,), can
`be designated as a translated delta function.
`It is informative to state two other conditions; firstly,
`that external electric fields arising from sources other
`than in the loop must be includedin theleft side of Eq.
`(2); secondly, the algebraic sign of a point delta func-
`tion field is taken as positive in the left side of Eq. (2)
`if it arises from a voltage source whose polarity drives
`current counterclockwise, and negativeifit arises from
`a gap or impedancesince it would then be a voltage
`sink. The first condition can arise in various ways, such
`
`8
`In terms of the functions shown in Eq.(6):
`
`Zon = ™No
`
`5.5
`a=),
`
`(Qatl)
`n(ntl)
`
`Jn(Bob)AnBoe) [ pro)]
`
`(8)
`
`Multiplying both sides of Eq.(6) by cos@ and integrat-
`ing from o to 27:
`
`Vo
`+2,
`F
`
`Lin
`
`Where Z,, = the perfect conductor k mode impedance
`looking into the loop.
`Again, in terms of the functions shown in Eq.(6):
`
`Lea = To
`
`(2nt1)
`*
`n=k+1, 2n(nt1)
`A+B...
`oc
`
`(n-k)!
`(ntk)!
`
`(Bob) An( Bot) [pa®*(o) }
`
`+ To
`
`k
`yi
`Hr Ce ea
`In(ntl)
`(ekkyt
`In (Bob)Hy’ (Boe) [ paX(0}]
`n=k,
`A+2,...
`OF Zee = Leste + Zeer
`
`z
`
`(10)
`
`(11)
`
`as a plane wavefrom a distance source whicharrives at
`the loop, or a mutually coupled field from a nearby
`source. The differentiation implied here between a 30
`distant source and a nearbysourceis that there is mutu-
`tal interaction among nearby sources, but negligible
`interaction with distant sources.
`Continuing the solution for the single-turn loop to
`determine the current coefficients,
`let us assume a 35
`delta function generator with voltage V, is applied at
`@=0° as shownin FIG. 1. Then Eq. (2) becomes:
`
`Eo re tte
`
`
`¢
`
`erty
`TM
`
`V, 8(b) — ZN) = 0
`
`(5)
`
`40
`
`Multiplying both sides of Eq.(6) by sin k¢@ and integrat-
`ing from o to 27:
`
`b, =o (because the delta function
`picks the value of sin kd
`at ¢d=0°
`
`(12)
`
`The external tangential electric field at rc due to
`‘current flow can be rewritten in general, as:
`wo
`.
`a.
`z
`TeeZa [ay coskptb, sinkb]Zee
`
`Ed (¢)=—-
`
`(13)
`
`Substituting the values of E,at r=c, 6=90° from the
`TE and TM wavefamilies:
`
`wherethe superscript I refers to the external induced
`field
`
`Yabo) aS egmtett) “Gntege In(BobdAM(Bee)
`
`"
`
`(Qnt1)
`
`(n-k)!s
`
`-
`
`»
`
`.
`
`AtBe.
`
`d
`
`fag [reo]
`
`6=90°
`
`;
`
`[a,coskd+bysinkd]
`
`(6)
`
`wo
`.
`A 2nt1)
`(nok)!
`.,
`2
`.
`+ a ey > “TrT) Geb)! Je'(Bob)Hn'(Boc) [rstco] [a,cosko+b,sinkd]
`A+,
`
`+ eZ! fa,+
`
` (aycoskd + bysinkd)]
`%
`k=1,2,...
`
`The impedance looking into the loop (and admit-
`tance) is:
`
`The use of orthogonality relations results in a determi-
`nation of the current coefficients in terms of known
`values. Thus, integrating around the loop over ¢ froma 6g
`to 22, da) is found immediately as:
`
`V,ee
`ae Fate,
`
`(7)
`
`65
`
`or Yan, =
`
`V,
`Mo)
`
`_
`Zin ™
`
`I(o)
`V, =(Y¥,+
`
`°
`Far
`
`Yx]
`
`where Z,, = the impedancelooking into the loop for the
`zero mode, k =o, for a perfect conductor;and Z, = the
`total internal impedance aroundthe loop.
`
`where Y, =
`
`1
`‘os
`“ZF TZ,
`
`1
`
`|
`
`
`
`(14)
`
`(15)
`
`

`

`9
`y, = —
`‘
`Zin + Zz
`a)
`
`3,956,751
`
`16)
`
`(
`
`5
`
`10
`replaced by [a,cosk@n+b,sinkha]. where each ¢a is
`given: Also, in addition to an equal number of equa-
`tions, similar to Eq.(19), there will be an equal number
`of equations for b,/V,. Substitution of three equations
`for a;/Vo and b,/V, in each new Eq.(17) and simulta-
`voltae solution of all the new Eq.(17)’s will yield all the
`The final step in the solution of a single-turn loop is to
`incorporate the effect of one or more lumped imped-_-V0l#BeTatlos.
`ances inserted in the loop periphery. To showthis, let
`BASIC SOLUTION FOR ELECTROMAGNETIC
`us take a specific example, an impedance inserted at 1g
`SENSING BY A THINWIRE ONE-TURN LOOP
`wette the left side of Eq.(5) would have an
`A. GEOMETRIC RELATIONS
`‘
`FIG. 1 illustrates the geometry for a loop antenna
`lying in the X-Y plane of a spherical co-ordinate sys-
`15 tem. The voltage. V,, at 6=0° is now, however, the
`voltage developed across a load impedancerather than
`a driving voltage.
`The first geometric relation needed is the angle be-
`tween two finite length lines which do not meet. This
`The negative sign in the first relation of Eq.(17) ac- 29 angle is defined as that. between twointersecting lines
`counts for the fact that the voltage across the imped-_Parallel to the givenlines and having the samepositive
`ance Zigo is equivalent to a delta function sink. The
`directions in terms of their direction cosines. Thus, for
`left side of Eq.(6) would now become:
`example,
`in FIG. 1, if r is a line through the origin
`[V08(b) — Vi1a026 (— 180°) ] Eq. (6) modified
`which representsa parallel line which originally did not
`Applying orthogonality relations as before results inthe 25 Pass throughtheorigin, and b represents anothersimi-
`following:
`lar line, the angle between them is given by:
`costs = cosa, cosa,tcosB cosB, +cosy,cosy,
`
`1
`= View 8(— 180°) oF Er 546
`
`also Viao:= Zig0°
`
`1(180°)
`
`(17)
`
`oF Vigo: Yao
`
`= 1(180°)
`
`(21)
`
`or
`
`ay = ¥4{V.— (-1)*Viso]
`b= 0
`—- Yuin= Ye [cn ve|
`
`®V woe
`
`A=0,1,...°
`
`(18)
`
`(19)
`
`30
`
`Note that the mode admittances, Y,in, are now given in
`terms of the mode admittances, Y;, with no inserted 35
`impedance, andthe ratio of voltage across the inserted
`impedanceto the driving voltage, Viso/Vo. From Eq.
`(17):
`
`;
`Where
`a, = angle a line makes with the +X axis
`_
`:
`.
`.
`Bn = angle a line makes with the +Y axis
`Yn = angle a line makes with the +Z axis
`Thedirection cosines of the line r, in terms of the coor-
`dinate angles @ and ¢ oftheline are:
`
`cose, = sin8 cosé
`cos 8, = sind sing
`cosy, = cosé
`
`(22)
`
`co
`ewe = OE CE oe
`V
`180°
`V,
`°
`A=0,1,2,...
`°
`
`a7)
`
`Substituting Eq. (19) in Eq. (17) and solving for
`VisoelVo:
`
`ao
`
`View
`Ve
`
`=
`
`k=0,1,.2_
`Yisoe +
`
`(=*
`a
`$
`k=0,1,2
`
`Y,
`
`(20)
`
`Since the right side of Eq.(20) consists of known admit-
`tance, the voltage ratio is now a known quality. Substi-
`tution in Eq.(19) and summation over k yields the new
`value for the input admittance with an impedancein-
`serted at @ = 180°.
`The solution for an impedance inserted at any ¢ or
`for any numberof impedancesis easily obtained. Thus,
`
`40 Given a line in space which does not pass through the
`Origin, its direction cosines are foundbyfirst translat-
`.
`2
`.
`ing the line parallel to itself so as to. pass through the
`origin, and then calculating the polar and azimuthal
`angles, 6 and ¢, of the translated line. Now consider the
`45 two systemsof three vectors in FIG. 1, (E,, Ee, Es)
`or (H;, Hs», Hs). For either one,
`the r ‘direction
`vectorlies along r, the @ direction vector is normal to r
`andlies in the plane described by r and the Z axis, the
`¢ direction vector is normal to r and also normalto the
`50 plane described by r and the Z axis. This plane is usu-
`ally called the plane of incidence. The three vectors are
`always in the direction of increasing value of the coor-
`dinates, r, 0, d. They form a CCW system. It is required
`to find the direction cosines of the three vectors when
`55 the radius vector to the point P may lie anywhere in 47
`steradians.
`Analyzing the eight cubical spaces in which the ra-
`dius vector maylie, the direction cosines of the three
`vectors are found to be the following:
`
`cosaz =sindcosd
`cosBz,=sindsind
`cOSyg,.—Ccosé
`
`=cosécosd
`COStg,
`=cosésind
`cosBrg
`Cosyeg = —sind
`
`COSae¢ = —sind
`cospeg = cosd
`cosyag = 0
`
`(23)
`
`for a numberof inserted impedancesat various points,
`a, there will be an equal numberof equationssimilar to
`Eq.(17) but with the term on the right side, (—1)*Ak,
`
`The @ and @ coordinate values above are those for
`the radius vector,r.
`
`

`

`3,956,751
`
`11
`A second geometric relation needed is the equation
`for the normal distance from a plane in space to any
`point in space. If we allow r, to represent the normal to
`the plane from an origin of coordinates, then the nor-
`mal distance from the plane to any point (x,y,z) is:
`deeyziTon(XCOSArg + YCOS ByotZCOSYrq)
`(24)
`
`B. SOLUTION FOR SINGLE PLANE WAVE
`INCIDENT ON LOOP
`
`In FIG. 1, assume a plane wave incident on the loop.
`
`12
`Substituting Eqs. (22) and (28) in (24):
`duryi tome sin A.cos [bo- 7]
`
`(29)
`
`5
`
`The phase delay is expressed by e? B 9 4(x,y,z). Since
`the phase referenceis taken at the origin, it is necessary
`to substract Bor, from Bodisy,2).
`_
`Hence, the phase factor required is:
`e * By al 1Bolo = eBo «nn 8,
`cox Py - $, ,
`
`(30)
`
`Thus, the tangential projections of the incidentfields,
`including the variation in phase across the loop,are:
`
`E.by
`
`Ed,
`
`con by - o;! ]
`[Ene 1B. #98,
`cos( doo)
`[Feta Bo rm Be do -)) sin(@.—;)cos6,
`
`(31)
`
`(32)
`
`Let r be the radius vector r,, normal to the plane wave
`front. Let 6-0, and @ = ¢, specify the orientation of r,
`relative to the spherical coordinate system. Let E ¢
`and E g of FIG. 1 be twoelectric fields, E,¢%1
`and
`E,e/*?
`respectively which are contained in the plane
`wave front. The phase factor for each, a, and ag,
`is
`considered to be the value of the phase of each at the
`origin of the loop coordinates. The two different phase
`factors imply an elliptically polarized wave, the most
`general case in plane waves. The Ey field is normal to
`the plane of incidence defined by the r,;—Z axis vectors,
`while the E, field is parallel to the plane of incidence
`and lies in it. The variation in magnitude of eachfield
`as each passes across the loop can be considered to be
`negligible. The change in phase, however, in traversing
`the loop, is not.
`The incident plane wave causes a flow of current in
`the loop, assumed in the CCW direction. Hence, a load,
`Z,, placed across the V, terminals at @ = O°will exhibit
`a voltage. It is required to find this signal, V;.
`Thedirection cosines of Eq.(23) will be used in con-
`junction with the direction cosines of the tangent line
`to any point on the loop to find the tangential projec-
`tion of the incident fields on the loop. If ¢, is the azi-
`muthal coordinate of the radius vector to any point on
`the loop, (@-=90°), then the direction cosines of the
`tangent are:
`cosa, = —sind, cos B=cosd, cosy= 0
`
`(25)
`
`25
`
`30
`
`35
`
`40
`
`45
`
`Refer now to the boundary condition equation, Eq. (5)
`which should now be rewritten as follows:
`Eg'+E¢g ert —(1/c) V, 86) — Z(G) = 0
`
`(33)
`
`oO
`where
`E g ‘= external field due to current flow
`E ° = external field incident on loop
`—1/c V, 8(¢) = point field in loop at @ = 0 due to
`current in load
`Z'‘I(@) = internal electric field due to finite conduc-
`tivity
`Substitution of Eq. (13) for Eg / and applying orthog-
`onality relations and integration as before:
`
`—v,+c
`
`fon E g cos kbd
`ze
`
`Ke=041,2... ©
`
`(34)
`
`c
`
`[27 Bg
`Jo
`
`od
`13
`z
`
`“sin kbdd
`
`K=1,2,.... ©
`
`(35)
`
`a=
`
`b=
`
`Now
`
`VY =Mo)={[at
`
`a
`=
`
`al
`
`(36)
`
`Substituting from Eq.(34):
`
`WY, + Vi
`
`a
`
`z
`
`k=0,1-
`
`t
`
`[+] =c
`
`al
`
`a
`
`x
`
`K
`k=0,1,...
`
`f Qa
`°
`
`coskbEdh stldd
`Zi,
`
`(37)
`
`
`
`Using Eqs. (21), (23), and (25), and neglecting phase
`change momentarily, the tangential projections of the
`incident fields are:
`
`55
`
`From Eq.(14), the second term on the left is V/Yin
`where Yj, is the loop input admittance. Hence:
`Ss
`:
`[ Y,
`rr"
`|
`(26)
`Edy =Ey!™ cosié.-11
`2 4=0.1,..L7 coskdEqh"Td0 a g
`
`
`
`(27)
`V+ Yn
`n=
`(38)
`Ed, = Ep 72 cosO,sin [¢.— $7]
`60
`
`where S = circumference of loop=27c
`Note that in Eq.(34), if E @ ** is zero, summing both
`sides over k would yield:
`
`(39)
`
`iT
`
`o")
`
`Ya =
`
`To incorporate phase change, assume that the phase
`reference pointis at the center of the loop. The (x,y,z)
`coordinates of any point on the loop are:
`
`x=c
`yc
`z=0
`
`cos@;
`sing,
`
`65
`
`(28)
`
`

`

`13
`This states that the input impedanceto the loop is the
`negative ratio of V,/T(0°). Since V; was taken as a
`voltage drop, this result is consistent.
`As
`in the single-turn loop solution previously
`derived,
`the effect of other
`inserted impedances
`anywhere in the loop can be easily found by adding
`terms of the type —1/c V4ad(¢@—¢a) to the left side
`of Eq.(33) and following the same solution procedure
`as previously.
`The next step in the solutionis to substitute the value
`for the incident field, which is the sum of Eq.(31) and
`Eq.(32),
`in Eq.(38). The exponential factor in the
`incident field can be expanded using the formula:
`ao
`
`3,956,751
`
`Vor = —japgH
`
`14
`
`(45)
`
`In general, the signal voltage is dependent upon the
`mode admittances (impedances). Hence,it is useful to
`know their characteristics relative to each other and
`relative to frequency. While I have developed and pub-
`lished curves showing these impedances over an elec-
`tric circumferential range up to s/A equal to 2.4, | am
`mainly interested here in somewhat smallelectrical
`loops, 0.2A in circumference or less. The perfect con-
`ductor mode impedances for such size loops are the
`following:
`
`a z
`
`
`
`(40)
`ebteore =],(M)+2 (-1)VoglM)cos2get+2j==
`(-1)ager (M)cos(2¢+1 Je
`q=1
`q=0
`
`1. The zero mode impedance is basically a small
`where M = B.csing,
`inductive reactance, hence a large admittance.
`20
`cos € = cos(¢,—¢,)
`Impedance increases with frequency.
`2. The unity mode impedanceis mainlyafairly large
`JM) = cylindrical Bessel function
`capactive reactance, hence a small admittance.
`The impedance decreases with frequency. How-
`ever, overthe range up to 0.2, its magnitudeisstill
`much larger than that of the zero mode.
`3. The higher order mode impedances are increas-
`ingly much larger capacitive reactances than that
`of the unity mode. Over the range up to 0.2A, the
`:
`:
`:
`admittances of the modes higher than the unity one
`can be neglected without introducing significant
`errors.
`Based on the above considerations, Eq.(43) will be
`expandedin the zero and unity modes.In addition, for
`x less than 0.2A, J,,(x) is closely equal to X"/n!2" and:
`
`where J,’ = derivative of Jg with respect to x
`The final equation is the following:
`
`35
`
`The details of the integration of the expanded form of
`Eq.(38) which now incorporates a double summation, 25
`over k and q, are lengthy but simple. Consolidation of
`the final result is achieved by the use of the following
`recursion relations:
`Jq-1(X)Joa
`(x) = 2S q
`oor 7
`Jana (2)4Jga(2) = 24 Jelx)
`*
`
`41
`“y
`(42) 30
`
`a
`
`ZL
`k=0
`
`Vy = —js
`
`;
`ai
`By
`(—1)42Y,[EycoskdoJs' (Bocsing,) d@ +E,sinkd,
`Y,+ Yu
`
`—Keosb,
`"
`Bocsin@,
`
`i
`:
`Ji (Bocsin®,) ei]
`
`(43)
`
`oo,
`.
`:
`=.
`—B,c
`sind,
`
`The open-circuit voltage developed is obtained by al- Jax) = Six) => = ioste
`lowing Y, to equal zero. The ratio of V; to Vog is:
`:
`(46)
`If
`x—J,
`¢
`Sind,
`45 Nx) = EERO LyBecsinter
`
`Vi
`Yin
`z
`Ve YitY¥um
`oo
`LZ+Zy
`(44)
`Then Eq.(43) becomes:
`
`_.
`mai
`
`Ss
`9
`
`{ [YoBoc sind,—j¥icosdo]Exe’1 —j¥,sing,cos0.Fye/~2
`Yt Yot+ Yi
`
`(47)
`
`The equivalent circuit is a simple series circuit repre-
`sented by a generator of voltage V,, with an internal 55
`impedanceof Z;,, in series with a load Z,. Any other
`external fields present can be accounted for by adding
`them, with their specific values of 0,, @, and a, to the
`right side of Eq.(43).
`In the case of a very small electrical size loop in 60
`which only the k=0 mode need