-
`
`-------,
`
`RADIO SCIENCE Journal of Research NBS/USNC- URSI
`Vol. 69D, No.7, July 1965
`
`Theory of Coil Antennas
`Trilochan Fadhi
`Division of Engineering and Applied Physics, Harvard University, Cambridge, Mass.
`(Received October 21, 1964; revised December 12, 1964)
`
`In this paper, a method is presented by which the distribution of current on so me structurally
`simple coil or multi turn loop antennas may be obtained. The input admittances of unshielded and
`shielded coils are determined and their operation as receiving elements is considered.
`
`1. Introduction
`The high-frequency behavior of thin circular loop
`antennas has been studied in the past by se veral in(cid:173)
`It is shown in this paper
`ves tigators [Wu, 1962J.
`how res ults pre viously obtained for single-turn loop
`antennas can be applied to give the admittance and
`distribution of current for coils that possess sy mm etry
`in the sense that all turn s of the coil are equivalent.
`Such symmetry obtains if at any section of the coil
`the wires appear at the corners of a regular polygo n,
`and further if the main diagonal of this polygon is
`much smaller than both the radius of the coil and the
`wavelength in air.
`One advantage of the coil over the si ngle-turn loop
`antenna is that at low frequencies, the output voltage
`is direc tly proportional to the numb er of turn s of the
`coil whe n it is used as a receiving antenna. Also its
`higher impedance may make tuning or matchin g a
`simpler problem . For these reasons coils are widely
`used in direction finders and radio receivers. A recent
`application of coils is in th e production of very high rf
`magnetic fields . The prese nt theory may be used to
`calculate the voltage distribution on th e coil and hence
`estimate the maximum magne tic fi eld obtainable before
`electrical breakdown occurs.
`In practice, the detailed structure of the coil may
`not permit rigorous application of the following analysis
`but in most cases it should be possible to obtain semi(cid:173)
`quantitative estimates.
`Instead of solv·
`In brief, the method is as follows.
`ing directly for the current on each of the N turns of
`th e coil, it is found more convenient to regard the
`current on each turn as the superposition of currents
`in N phase sequences.
`In the sequence for which the
`phase difference between the currents on any two
`turns is zero, the current distribution is the same as
`that on a single-turn loop of appropriate equivalent
`cross-sectional size. Other sequences are nonradiat(cid:173)
`ing and behave essentially like the TEM mode on a
`two-wire line. The current in e ach sequence is, there(cid:173)
`fore, known or can be found, and the current and input
`admittance of the coil determined by summing up the
`co ntributions from each phase sequence.
`
`\
`/
`
`2. Current Sequences
`Let N be th e number of turn s on th e coil. A point
`on the coil is de noted by X whi ch is its a ngular distance
`from some fixed point on th e coil , meas ure d along th e
`co ndu ctor. The c urrent J(X) across th e sec ti on of the
`co ndu ctor at X must sati sfy th e periodicity co nditi on
`I(X) = l (X + 21TN).
`
`Hence, J(X) may be written in a Fouri er seri es:
`J(X) = 2: Ame-jmX/N.
`
`00
`
`m = - :x>
`
`To display this as a sum of phase seq uences put
`
`m = k+nN;
`
`(k=O,I,2, ... ,N-I;
`
`n = O, ±I,±2, . .. , ±oo).
`
`An eq uival e nt form for J(X) is, therefore,
`J(X) = 2: e-jkx/N 2: A~e-jnx.
`
`.\' - 1
`
`Xl
`
`k = O
`
`n = - oo
`
`The (1 + i)th turn of th e coil is defin ed by
`
`He nce the current on the (1 + i)th turn is
`
`where
`
`N-l
`
`= 2: e-21Tijk/NJk(t/J),
`
`k =O
`
`Jk(t/J) = e-jkw/N 2: A;'e-jllw.
`'"
`
`n = - x
`
`(1)
`
`Jk(t/J) IS called the k-sequence current. When a pure
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`k-sequence current exists on the coil, it IS implied
`that the current on the (1 + i)th turn is
`
`f7(1jJ) = e-2rrijkINJk(1jJ).
`
`It foliows from (2) that
`
`N - I
`J'to/1jJ) == 2:J}(1jJ) = NJk(1jJ)8kO;
`i=O
`
`(2)
`
`(3)
`
`that is, in each sequence k "'" 0, the total current across
`a section of the coil is zero.
`
`3. Driving Voltage Sequences
`Suppose a voltage generator of strength Vi is intro(cid:173)
`duced into each of the N turns at 1jJ = 0 (fig. 2). Define
`N sequence voltages Vk by the set of equations
`
`N-I
`Vi=L e-2rrijkINVk; (i=O, 1, .
`k=O
`
`.
`
`. , N-1).
`
`(4)
`
`The solution of (4) for Vk is
`
`(5)
`
`Consider first the distribution of current induced by
`a zero-sequence voltage. By suitably altering the
`connections at the N driving points, the problem of the
`coil reduces to that of N identical, closely coupled,
`circular antennas driven symmetrically, i.e., by equal
`voltages in phase. The change in connections leaves
`It is shown in
`the distribution of current unaffected.
`the literature [King and Harrison, 1965] that for two
`identical, closely coupled loop antennas driven in
`zero-phase sequence, the distribution of current is the
`same as that on an isolated loop of the same diameter,
`but with the wire of radius a replaced by wire of radius
`Ii = vi (ad), where d is the separation. A simple ex(cid:173)
`tension of this result to the present case shows that
`the equivalent loop is one with the same mean diameter
`as the coil but made of wire of radius a given by
`
`where di is the distance of the (1 + i)th turn from the
`first turn. The current JhlOP(1jJ) and admittance yOOP
`for the equivalent loop have been evaluated for small
`a in an earlier reference [Wu, 1962], and numerical
`results are available [King, Harrison, and Tingley,
`1963].
`In terms of these known quantities the zero(cid:173)
`sequence current on the coil is, therefore,
`
`For the case where only one source is present (say in
`the first turn), this simplifies to
`
`Vk = V/N, where Vi = V8iO.
`
`(Sa)
`
`4. Phase Sequence Input Admittances
`Great simplification of the problem of coil antennas
`results if it is assumed that there is sufficient symmetry
`to ensure that if the coil is driven by a k-sequence
`voltage, the currents, too, form the same sequence.
`Coils that approximately satisfy this condition may be
`wound by making sure that at any section, the turns
`of the coil appear at the corners of a regular polygon.
`Further, since all turns should be as nearly as possible
`of the same length (physically as well as electrically),
`the separation between turns should be small com(cid:173)
`pared to both the radius of the coil and the wavelength
`corresponding to the frequency of operation. A four(cid:173)
`turn coil is shown in figure 1; for clarity the separation
`between turns has been exaggerated.
`
`v
`
`The zero-sequence input admittance IS defined by
`
`Hence
`
`1
`yO="N YooP.
`
`(6)
`
`For all higher sequences, radiation from the coil
`can be neglected as a consequence of (3) and the condi(cid:173)
`tion that the separation between any pair of turns be
`mu ch smaller than a wavelength. Now the theory of a
`straight, infinitely long, multiwire transmission-line
`
`FIGURE 1. A four turn coil in which the conductors appear at the
`corners of a square in every cross section.
`
`FIGU RE 2. Schematic view of coil.
`-
`One turn is drawn with a heavy line.
`
`998
`
`'I
`I
`j
`
`I
`
`1 i
`
`(
`1
`~
`J
`\
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`[Tai, 1948] shows that when it is driven by a k(cid:173)
`seque nce voltage (k 01= 0), a TEM wave propagates
`Propagation
`along the line with the velocity of light-
`along th e line is essen tiall y unaffected eve n if the lin e
`is curved, provided th e radiu s of curvature is mu ch
`th an
`the separation between condu cto rs .
`large r
`He nce, the k-sequence current /k(ifJ) and (in ana logous
`notation) the k-sequence scalar potential ¢'·(ifJ) obey
`(fo r ifJ 01= 0) the eq uations
`
`~ d2¢k + f32,1,.k = 0
`R2 difJ2
`'f'
`,
`
`(7)
`
`(8)
`
`a nd
`
`where
`
`R = mean radius of the coil,
`
`Zk = characteristic
`re(cid:173)
`impedance
`i. e.,
`impedan ce ,
`quired in e ac h arm of a s tar-connec ted load th at
`matches the transmission-line when it is exc ited
`by a k-sequence voltage,
`
`and
`
`f3 = phase constant of th e em wave.
`
`At ifJ = 0 there is a discontinuity in scalar pote ntial
`(see (Sa))
`
`Furth er, (1) gives
`
`and, a nalogously
`
`(9)
`
`(10)
`
`(11)
`
`The solution of (7) and (8) s ubj ect to (9), (10), and (11)
`is obtained in the appendix _ The k- seque nce input
`admittance is shown to be
`
`.
`si n 21Tf3R
`k = /k(O) _ ~
`. )'
`)
`(k
`(k
`VA" - 4Zk
`Y -
`sin 1T N + f3R
`sin 1T N- f3R
`
`(k 01= 0).
`
`(1 2)
`
`Hence
`
`_ yOOI)
`
`. sin 21Tf3R
`Yin - N2 +,
`4N
`
`.\" - 1
`
`1
`
`)-
`(k
`(k)
`X L
`k = 1 Zk sin 1T N+ f3R sin 1T N- f3R
`
`(1 3)
`
`It is interesting to note that for Nf3R < < 1, (1 3)
`r e duces to the familiar low-frequency formula ,
`Y in = yOOP / N2 _ In fact, as long as the coil is perfectly
`conducting and the medium surrounding it is lossless ,
`the input conductance is always I/N2 times th e con(cid:173)
`ductance of the equivalent loop, irrespective of the
`size of the coil.
`Another well-known feature that follows from (13)
`is the antiresonance associated with the di stributed
`capacitance betwee n th e several pairs of turn s. At
`low frequenc ies, the input admittan ces of all se(cid:173)
`que nc es othe r th an th e zero-sequ e nce are c apacitive,
`while the zero-sequ e nce input admitt a nce is inductive .
`The net s usceptance becomes zero at a frequency that
`is quite low . For some typic al cases c alculated below,
`antiresonance occ urs at Nf3R ~ 0.2;
`i.e., th e total
`le ngth of wire used in the coil is only a fifth of a
`wa vele ngth.
`Calculated valu es of s usce ptance for coils of 2, 3,
`and 6 turn s are s hown in fi gure 3. For eac h coil th e
`di s tance betwee n adj ace nt turn s has bee n taken as
`twi ce the di a me ter of th e wire used. The actu al size
`of the wire is c hosen s uc h th at 2 In 21TR/"a = 10.
`In
`order to s how the reso nances clearly, the scale for
`th e ordinate in fi gure 3 is c hose n to be proporti onal
`to th e arc tan ge nt of the s usce pta nce (multiplied b y a
`It will be noti ced th at the be havior of the
`consta nt).
`input s usce ptan ce rese mbles that of a s hort-circuit ed
`line quite closely exce pt wh ere f3R is
`tran s mission
`near an intege r. At points of antiresonance or zero
`susceptance, the radiati on fro m th e coil is signific ant.
`Howe ver, wh ere the s usce ptance beco mes infinite,
`one of th e tran s mission-lin e modes effective ly s hort
`circuits all others a nd radiation is a minimum . This
`th e co ndi tion of reso nan ce . Reso nance occ urs
`is
`f3R = n + k/N ; (k = I, 2 , . .. , N-I ;
`wh e n e ve r
`n = O, 1, 2, _ .. , 00).
`
`Eq uati ons (6) and (12) give the input admittance for
`eve ry seq uence, k = 0, 1, 2, ... , N-l.
`It is now a simple matter to calculate the admit(cid:173)
`tance Y in presented to the source at ifJ = 0 on the first
`turn:
`
`?
`
`tV- I
`Yin=/O(O)/V = Ly')N.
`k =O
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`5. Shielded Coil Antenna
`A method almost identical to that employed above
`may be used to analyze the shielded coil or multi turn
`loop antenna.
`In its most common form, a shielded
`coil antenna consists of a coil threaded through a con(cid:173)
`ducting tube that is bent into the form of a ring. The
`ends of the tube are brought close together but do not
`meet. Here too, the method only applies to coils of
`the kind described in earlier sections, enclosed in
`shields
`that do not disturb
`their symmetry. An
`idealized configuration is considered, in which the
`generator (or load) in series with the coil is itself
`within the shield. For convenience, the gap in the
`shield is taken at 1/1 = 7r, (where 1/1 = 0 is the position
`of the generator).
`It is assumed throughout that the
`shield is thin and perfectly conducting, and that the
`diameter of its cross section is much smaller than a
`wavelength.
`The current on the coil and, similarly, the scalar
`potential, may be decomposed into sequences exactly
`as in section 2. For any k-sequence, k =P 0, (7), (8),
`(9), (10), and (11) hold because, as a consequence of
`(3), the gap in the shield at 1/1 = 7r does not affect its
`behavior. Hence, for k =P 0, the k-sequence input
`admittance is given by (12). Of course, the charac·
`teristic impedance Zk will now be different because
`of the presence of the shield.
`The current Ish (1/1) across a section of the shield at
`tfJ may be split into two parts:
`
`Let cpsh(tfJ) be the scalar potential on the shield. The
`zero-sequence potential difference between the coil
`and the shield is
`
`10(1/1) and 1>diff(I/I) together form a nonradiating, trans·
`mission-line type of wave, that exists within the shield.
`On the other hand, the part l"ad(I/I) of the current on the
`shield radiates like the current on a loop of the same
`shape and size as the shield. Hence, in terms of the
`admittance of this equivalent loop,
`
`The loop mode and the transmission· line mode couple
`through the gap at 1/1 = 7r where the following equations
`must hold:
`
`and
`
`Hence
`
`Thus, in effect, the loop mode places an impedance
`N/yiOOP in series with each turn of the coil at 1/1 = 7r.
`
`The zero·sequence input admittance is now readily
`found to be
`
`_ 1 2Z0y loop + jN tan 7r{3R
`yO - 2Zo N + 2Z0yiooP tan 7r{3R
`
`(14)
`
`Using (12) and (14), the input admittance is obtained
`from
`
`N-I
`yn=2: yk/N.
`k =O
`Formulas for Zk to be used in calculating y' are given
`in the appendix.
`
`6. Receiving Cbil Antenna
`The unshielded coil is considered first. Assume
`that the effective length [King and Harrison, 1965]
`of the equivalent loop is known and that the coil is
`loaded by an impedance R at tfJ = 0 on the first turn.
`From the definition of effective length it follows that
`if all turns of the coil were open·circuited at tfJ = 0,
`the resulting potential difference across the open(cid:173)
`circuit would be the same for all turns and would be
`given by
`
`V'nd =- 2he . Einc
`
`where
`
`Einc = incident electric field
`
`and
`
`(15)
`
`2he = effective height of the equivalent loop; h e is a
`function of the direction of propagation of the
`incident field and its polarization.
`
`In normal operation as a receiving element, the
`voltage sequences must be given by
`
`N-l
`VI' = - RIo(O)/N = - R 2: Ik(O)/N.
`k =O
`
`(16)
`
`~ I
`
`But, in terms of the current sequences Ik(O) ,
`
`VI' = V'nd 8kO + 1" (0)/ yk.
`
`Using this and (16) to solve for lIk(O) yields
`
`% Ik(O)= - R +\/yn (;: V'nd) .
`
`The Thevenin equi valent of the coil at its output ter(cid:173)
`minals is, therefore, a voltage generator of strength
`- yOV'nc/yn in series with an impedance l/yin. Alter·
`natively, this may be regarded as a current generator
`of strength -yOV'nd in parallel with an admittance, yn.
`The derivation from this simple picture, of the low(cid:173)
`frequency result that the output voltage of a coil is
`proportional to the number of turns, is interesting.
`First it will be noted that since yO = yiOOP / N, the strength
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`of the current generator decreases as N is increased
`and is in fact roughly proportional to liN. But at low
`frequencies, yn is proportional to 1/N2. Hence, at
`low frequencies, the open·circuit output voltage is
`proportional to the number of turns.
`The equivalent circuit of a shielded receiving coil
`may be obtained by a similar method. Here, a simpler
`method will be used, which rests on the fact that the
`internal admittance of the shielded loop, viewed from
`its terminals, is the same whether it is being used as a
`transmitter or a receiver. The strength of the current
`generator in parallel with the internal admittance is
`found by calculating the current that flows through the
`output terminals when they are short circuited.
`In
`this condition, only zero-sequence currents are ex(cid:173)
`cited. Using elementary transmission-line theory, it
`can be shown that the strength of the equivalent cur·
`rent generator, when viewed from the terminals at
`t/J = 0 on the first turn, is
`
`N cos rr{3R + 2jy100PZo sin rr{3R
`Her e yOOP and ZO hav e the meaning they had in sec(cid:173)
`tion S, and 0 nd is the voltage induced across the gap
`of the equivalent loop, ope n-circuited at t/J = rr . Just
`as in the unshielded case (IS), one may write down
`0 nd in terms of an effective le ngth, but care should
`be taken to see that it refers to a load situate d at t/J = rr.
`7. Appendix
`The solution of the wave equation (7), subject to
`the boundary conditions (9) and (11) is
`cf>k(t/J)=! VI' ~ sin ~R(rr -i t/JI)
`2
`t/J
`SIn (3Rrr
`+ . cf>k(rr)
`[sin {3R(rr + t/J) + e27Tjk/N si n (3R(rr - t/J)].
`S In 2{3Rrr
`
`Equatio n (8) now gives
`
`[k(t/J)=.L {_ VI' cos (3R(rr - it/Ji )
`2
`si n (3Rrr
`Zk
`+ . cf>k(rr)
`[cos (3R (rr + t/J) - e27Tjk/N cos {3R(rr - t/J)]}.
`SIn 2{3Rrr
`
`(AI)
`
`This expression for [k(ljJ) m.ust satisfy the boundary
`co ndition (10). This is possible if and only if
`
`cf>k(rr) =- jVl'
`
`cos (3Rrr sin krr e- j7Tk/N
`N
`2rrk
`cos N- cos 2rr{3R
`
`(A2)
`
`sin 2rr{3R
`
`)'
`)
`(k
`(k
`sin rr "N+{3R sin rr ""N-{3R
`
`.
`
`The only quantity that remains to be c alculated is
`Zk. The formula (neglecting proximity) to be used in
`calc ulating the sequence admittance for the unshielde d
`coil is avialable [Tai, 1948] for k= 1. The generaliza(cid:173)
`tion to other values of k is not difficult and yields
`
`[ d
`27Tik
`Zk=60 In ~- 2 ~ cos N
`
`S
`
`. rri]
`In SIn N ohms. (A4)
`
`For the shielded coil, the formula to be used is (for
`a« diN, D-d)
`
`d2)
`'.!.(1-d2/D2)
`[
`k -
`Z -60 Ina(l+d2/D2)+N8koln2d 1+D2
`
`R(
`
`S
`
`-2 L cos--In-sm- ohms.
`2rrik 1. rri]
`N
`N
`Lli
`
`i= 1
`
`(AS)
`
`In (A4) and (AS),
`
`d = diameter of the circle in which th e regular polygon
`defin ed by the co nduc tors may be inscribed,
`a = radius of each co ndu ctor,
`5 = N/2 or (N - 1)/2, whichever is an integer,
`D = diam eter of a cross sec tion of the shield, if any,
`
`and
`
`In these formulas , the proximity effec t is neglected.
`A study of the exact and approximate formulas for
`the case of the two-wire line s hows th at no serious
`error res ults if th e approxim ation is mad e for d/a ~ 4.
`Hence, in dealing with th e multiwire line , one ca n hope
`for reaso nable accuracy provided care is taken to use
`th e approximate formulas only whe n th e side of the
`polygo n i s a t leas t four tim es th e radi u s of th e
`co nductor.
`
`The author is grateful to R. W. P. King for his guid(cid:173)
`ance and kind encourage me nt.
`
`8. References
`King, R.W.P ., and C. W. Harrison, .Ir. (1965), Electro magnetic
`Radiation, ch. IX, sec. 7, and ch. X, sec. 2 (to be published).
`King, R. W . P. , C. W. Harrison, Jr., and D. G. Tingley (1963), The
`adm ittance of the bare circular loop antennas in a dissipative me(cid:173)
`dium, Sandia Corporation Monograph, SCR-{)74.
`Tai , C. T . (1948), High.frequ ency polyphase transmiss ion-lines,
`Proc. IRE 36, 1370.
`Wu, T. T . (1962), Theory of the thin circ ul ar loop a ntenna, J. Mat h.
`Phys. 3 , 1301- 1304, and the references given th erein_
`
`Substitution from (A2) for cf>k(rr) in (AI) leads to
`,,k _ [k(O) _ L
`Vk - 4Z k
`
`.!
`
`-
`
`(k =i' 0).
`
`(A3)
`
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