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Publication numberUS3823403 A
Publication typeGrant
Publication dateJul 9, 1974
Filing dateJun 9, 1971
Priority dateJun 9, 1971
Publication numberUS 3823403 A, US 3823403A, US-A-3823403, US3823403 A, US3823403A
InventorsMunk B, Thiele G, Walter C
Original AssigneeUniv Ohio State Res Found
Export CitationBiBTeX, EndNote, RefMan
External Links: USPTO, USPTO Assignment, Espacenet
Multiturn loop antenna
US 3823403 A
The invention is a multiturn loop antenna of an efficient design suitable for use singly, in arrays, or for inductively exciting radiating currents on a structure - the surface and the elements radiating to achieve optimum system performance. The antenna may be fed balanced or unbalanced and its input impedance may be either capacitive or inductive. In a preferred embodiment for omnidirectional coverage the antenna is positioned in a dielectric or ferrite filled cavity; the antenna may be a single element for linear polarization or a pair of multiturn loops at right angles to each other and with 90 DEG phasing to effect circular polarization.
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Description  (OCR text may contain errors)

United States Patent 1191 Walter et al. l July 9, 1974 [54] MULTITURN LOOP ANTENNA 3,569,979 3/1971 Munk 6t al. 343/895 1751 Carlton H-Walier; Benedikt ifiiifii 551335 X$fi3i33133331 .:..1::::: 3335532 Munk; Gary A. Thiele, all of Columbus Primary Examiner-Eli Lieberman [73] Assignee: The Ohio State University Research Allomey, Ag or Firm-'C6Imam0 Kremblas &

Foundation, Columbus, Ga. ter [22] Filed: June 9, 1971 [57] ABSTRACT PP N05 151,228 The invention is a multiturn loop antenna of an efficient design suitable for use singly, in arrays, or for in- [52 US. Cl 343/708 343/718 343/744 ductively exciting radiating currents on a 7 343/745 343/787 343/846 343/895 the surface and the elements radiating to achieve opti- 51 Int. Cl. I niild 11/12 mum System -i The antenna may be fed 58 Field of Search 343/705 708 895 90s balanced unbalanced and its input impedance may 343/867 7 7 8 be either capacitive or inductive. In a preferred em- 1 bodiment for omnidirectional coverage the antenna is 6 References Cited positioned in a dielectric or ferrite filled cavity; the antenna may be a single element for linear polariza- UNITED STATES PATENTS tion or a pair of multitum loops at right angles to each gf al other and with 90 phasing to effect circular polarizangs 3,365.72! 1/1968 Bump... 343/856 3,440,542 4/[969 Gautney 343/788 21 Claims, 35 Drawing Figures PATENTEB JUL 9 sum "01 or 14 INVENTOR. CARLTON H. WALTER BENEDIKT Au. MUNK BY GARY A. THIELE Wgzkiw ATTORNEY PATENTEDJUL 91914 j 3,823,403

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CAPACITOR EFFICIENCY l3-TURN LOOP 8 0.4-- 8 O3 6 5 5 0.2-- E LI- 111 o.|

F W mammal azusom'r AuJIUNK GARY A.THIELE BY Gamma 116ml! 8" 30.4fm-

ATTORNEYS PAIENIEUJUL 91924 sum as or 14 75 f(MHz) FIG. 8



ATTORNEYS PATENTED 1m emu maria I4 20 22 2426 2a 30 FREQUENCY MHz v FIG.I2

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oo. 03 03 oov can 03 02. T o8 com 08 cans INENTORS CARLTON H. WALTER I "ENEDIKT AG. IIUNK BABY .THIELE BY commune .K-unlaa Sealer ATTORNEYS PAIENTEUM sum 3.823.403 7 FIGITA FIGIT INVENTORS CARLTON u. WALTER BENEDIKT Au. IIUNK BY GARY A.THIELE Gennamo .JKremL/ad 8" floater ATTORNEYS PATENTEUJUL 91914 3,823,403





ATTORNEYS GARY A.TH|ELE small electrically. This 1 MULTITURN LOOP ANTENNA BACKGROUND I Because of space limitations, manyantennas at very high frequency (VHF) and below must be relatively generally implies an undesirable impedance characteristic and low efficiency. The success of such an antenna system may depend upon the efficient induction of tangential H-fields on any nearby support structure; e.g., the excitation of currents on an aircraft structure in order to make use of a larger radiating surface. Currents on a conducting surface are associated with an external tangential magnetic field that is maximum at the conducting surface. Such a magnetic field can generally be established more effectively by a loop element than by a stub or dipole element. However, the conventional small single-tum loop antenna is often too inefficient to be used as the basic radiating element in a feasible antenna system at VHF and below. 1

SUMMARY OF THE INVENTION The invention relates to a small diameter loop antenna with an efficiency greatly enhanced over that of a conventional loop by appropriately increasing the number of turns to achieve a natural resonance of the antenna. That is, the antenna is in the form of a multi turn loop that is particularly well suited for use in the veryhigh frequency (VHF) bands and below. This antenna may be used singly or in an array. It is more effcient than a conventional small loop having the same loop area and it is especially well suited for flush mounting. By exciting sizable radiating currents over the supporting structure its efficiency is increased.

An antenna system comprising a pair of multitum loops at rightangles provides omnidirectional radiation with improved efficiency. The array may be positioned in an isolated manner, above a ground plane or some metallic structure, or in a cavity which may be airfilled 4 or loaded with a ferrite or a dielectric that shifts the natural resonances downward in frequency. The antenna is small and since it may be flush mounted is well suited for use in compact receivers and transceivers.

A detailed and mathematical analysis is made for balanced and unbalanced tuning for adaptation at frequencies where the input reactance is either inductive or capacitive.

OBJ ECTS Accordingly it is a principal object of the invention to provide an improved loop antenna particularly suited to HF and VHF operation but also well suited to operation at frequencies below HF.

Another object of the invention is to provide an antenna system which permits omnidirectional radiation.

DESCRIPTION or THE DRAWINGS FIG. 5 is a graphical illustration of a wire segment of length s uniform current density on its surface;

FIGS. 6A to 6D illustrate schematically an array of orthogonal multiturnantennas in accordance with the invention;

FIG. 7 is a graphical representation of the input impedance of the 13 turn one-twelfth scale model loop antenna with parallel capacitive tuning;

FIG. 8 is a graphical representation of the efficiency of a five turn loop having a first resonance at about MHz;

FIGS. 9(a through e) are schematic illustrations of a sample of acceptable electromagnetic feed arrangements for the balanced multitum loop antenna;

FIG. 10 is a pictorial illustration of a practical .an- I tenna in a cavity built in accordance with the present invention; a v FIG. 11 is a graphical illustration tance with dielectric loading; I

FIG. 12 is a graphical illustration of. the input, reactance with'dielectric loading; I

FIG. l3is a graphical illustration of the efficiency of the input resis- 0 with dielectric loading;

FIG. 14 is a graphical representation of the input reactance with a tuning capacitor between loop and ground;

FIG. 15 is a graphical representation of the input resistance with capacitor termination on ground side of antenna;

FIG. 16 is a graphical illustration of the input reactance with capacitor on ground side of antenna;

FIG. 17 is a schematic side illustration of a pair of multitum loop antennas;

FIG. 17A is another side view of FIG. 17;

FIG. 18 is an illustration of stagger tuning at the multitum loop antennas to achieve phase shift for circular polarization;

FIG. 19 is a graphical representation of the input resistance with a tuning-capacitor between loop and ground;

FIG. 20 is a graphical illustration of the measured impedance for the antenna of FIG. 1 fed unbalanced and according to the design as depicted in FIGS. 15 and 16;

FIG. 21 is a graphical illustrationofmeasured and computed input resistance of the :rnultitum loop antenna;

FIG. 22 is a graphical illustration of the measured and computer input reactance of the multitum loop antenna;

and radius a of the turns of FIG. 4'with an unbalanced feed as a function of frequency,

FIG. 24 is a graphical illustration of the input resistance of a multitum antenna showing effect of tapping off various turns; 1

FIG. 25 is a graphical representation of the input reactance showing effect of tapping off various turns; and

FIG. 26is a graphical illustration of the multitum loop antenna in a preferred embodiment with capacitor matching (C and capacitor tuning ((3,).

BRIEF DESCRIPTION OF THE DRAWINGS The multitum loop antenna of the present invention consists, in one fundamental configuration, of about a quarter wavelength or odd multiple of quarter wavelength of conductor coiled into two'or more turns and mounted in or over a ground plane or metallic structure as illustrated in-FIGS. 1 and 4, respectively. The antenna in this arrangement is fed with an unbalanced line and together with its image forms about a half wavelength of conductor at its lowest antiresonant frequency. Another fundamental configuration consists of the antenna with about a half wavelength, or odd multiples thereof, of conductor and fed with a balanced line as illustrated in FIG. 9. A basic characteristic which distinguishes the multiturn loop of the present inven-- tion is its operation in the region of a natural antiresonance which substantially increases the efficiency over that of conventional small loops.

By choosing the number of turns and tuning so that the loop element operates below the first resistance peak, the reactive part of the input impedance of the multiturn loop element may be made positive. In this case, the balanced loop element can be matched to a balanced power amplifier by a purely capacitive matching network. This system eliminates matching coils, a major source of loss in-conventional antenna systems.

The effect of parallel capacitive balanced tuning is illustrated in 'FIG. 7. FIG. 7 is a Smith chart plotof the terminal impedance of one-twelfth scale model of the multitum loop as shown in FIG. 1 with and without a 0.75 pf. capacitor connected across its terminals. As seen in FIG. 7 the loop alone resonates at about 190 MHz while the loop and capacitor combination resonates at about I29 MHz. Thus the resonance frequency of the loop can be greatly altered by a parallel tuning capacitor.

The relative bandwidth of an antenna is the interval of frequencies over which the antenna reactance is less thanthe antenna resistance, divided by the resonance frequency. From, FIG. 7 the bandwidth of the tuned and untuned one-twelfth scale model multitum loop is about 26.3 percent for the untuned loop and about I 1.5 percent when the loopis tuned to resonate at 129 MHz. The bandwidth of the actual model is slightly greater than that measured on the scale model, since the loss resistance is scaled down, relative to the radiation resistance and reactance.

The efficiency curve of the antenna using all its turns is similar in shape to the curve of FIG. 8. The first efficiency hump appears at approximately the lowest operating frequency, other humps occur at 3, 5, 7, 9, times this frequency, and the low efficiency points are at frequencies in between. However, by

switching out turns the humps are moved so that most efficient radiation can occur'at chosen frequencies. For example, assuming a IO to 1 operating band, the first two regions of high efficiency corresponding to L z M2 and 3M2, would be used, and each would be tuned over a 3 to I band by switching out turns on the loop antenna. Therefore, efficiency calculations for various numbers of turns switched out of the loop antenna are made to obtain information on the efficiency throughout the operating frequency band.

In the construction of a working embodiment the efficiency performance for the particular loop geometry is analyzed as noted above. The loop conductor size and configuration, i.e., tube or wire bundle, is then chosen for an optimum combination of loop efficiency, weight, size, and complexity. Several alternative ways of feeding the balanced multitum loop are within the teachings of. the preferred embodiment, certain of these are shown in FIGS. 9a through 9e.

It has been determined that radiating currents are excited on a ground plane by the antenna of the present invention with significant effects. In essence, thev excitation of currents on the ground plane doubles the effective radiation, i.e., it is equivalent to an image antenna. The effective antenna is shown schematically in FIG. 4 and analyzed below.

In a practical working embodiment the antenna might not be positioned directly above the ground plane but could be integrally formed into the ground plane as shown in FIG. 26. The height of the loops a a" is somewhat smaller than the depth of cavity 14;

and the diameter of loops a a" is somewhat smallerthan the width of the cavity 14. The feed terminal in this embodiment comprises a coaxial cable 15 more clearly shown in the equivalent schematic illustration of FIG..4 with the center conductor of the coaxial cable connected to one end of the loop. The other end of the loop together with the outer conductor of the coaxial cable is connected to the ground plane 10. The other end of the center conductor is coupled to a transmitter, receiver, or transceiver through a simple tuning capacitor for proper impedance matching at the operating frequencies. The cavity is a small metal box or metalcoated plastic box which is used as a protective housing and counterpoise and not as a resonant cavity. The structure may be flush mounted into a small counterpoise which can be made light-weight, and may be designed to house other components of the system. By operating the antenna slightly below its natural resonance, the inductive reactance is tuned out by a single capacitor thereby providing for a match to a transmission line into the transceiver. The efficiency (see analysis below) of such a multitum loop antenna is far greater than that of'a monopole of comparable height and may approach that of a quarter wave monopole (whip). The relative efficiency of the multitum loop compared to a monopole increases as the sizev (i.e., maximum linear dimension) of the monopole decreases below a quarter wavelength.

The antenna shown in FIG. 10 is mounted in an airfilled cavity 14. The cavity 14 is so small in terms of the wavelength that there are no cavity resonances and its electrical-effect on the antenna is relatively small.

To permit reduction of the antenna size, the cavity may be loaded with a dielectric or ferrite. That is, for a given antenna dielectric or ferrite loading shifts the natural resonances downward in frequency as graphievidenced by the curves of FIG. 13.

In general whether the loop is mounted in a cavity or on a supporting body such as transmitter, receiver or transceiver, if the permittivity or the permeability (or both) of the medium in which the loop is placed is increased, the effect will be to slow down the traveling wave currents on the loop and thus makethe loop appear electrically larger in size. The effect will be to decrease the frequency at which the resistance peaks occur. Thus dielectric loading provides a means of effectively reducing loop size for a given level of performance. Loading the loop with a permeable material such as ferrite would have a similar effect. The loading may take the form of a core for the loop or the loop may be partially or completely embedded in the material.

In a first practical application of the multitum loop antenna of the present invention, the antenna of FIG. 1 comprising the 'l 3 loops in a cavity was fitted directly into a conducting surface. In this way a flush mounted antenna is provided. 7

, Another practical embodiment of the multitum loop antenna is that shown in FIGS. 17 and 17A. The multiturn loop is constructed as a module readily attached to the shoulder position of a vest-type garment 21. Owing to the fact that the electromagnetic fields are tightly contained in close proximity to the antenna, relatively little ground plane or counterpoiseneed be provided around the antenna 19 and 20. Therefore, it is to he expected that thebody effects are minimal or that they can be easily compensated for by proper system design.

This configuration of two orthogonal multitum loops give omnidirectional coverage in the upper hemisphere and thus have no deep nulls to hinder communications no matter what the position of the person wearing the antenna. The omnidirectional characteristic of a pair of right angle antennas is describedbelow. The antenna is connected to a transceiver 23. The antenna-speakermicrophone package could be attached to the vest garment with heavy duty zippers designed to provide electrical contact between the antenna and the small counter-poise 10 around the antenna and also provide a physically secure low profile means of attachment.

With particular reference now to FIGS. 6, 6A, 6B and 6C, there is shown a first multitum loop and a second multiturn loop antenna positioned at right angles to each other.

Under ideal conditions where it is not disturbed by any nearby objects the multitum loop has essentially the pattern of a small loop or magnetic dipole. When the loop is mounted over a large ground plane the resulting pattern is essentially that of the loop and its image. For a small ground plane (including cavity mounted loops) the total pattern can be obtained from 'a superposition of the pattern of the loop and-that of the currents that the loop excites on the ground plane structure. In general the pattern will have aminimum value along the axis of the loop. This minimum ranges from a perfect null for the ideal loop case to something on the order of 10 db or less in the presence of support structure. However, the pattern minimum along the loop axis is effectively eliminated by using a pair of multitum loops mounted orthogonally (see FIG. 6) and with 90 phasing. ldeallythis can be pictured as a twoelement array of magnetic dipoles at right angles to one another. With 90 phasing, the resultant pattern is circularly polarized on the axis of the array and vertically polarized with omnidirectional pattern in the plane of the array for anarray in the horizontal plane.

The 90 phasing may be achievedby a quarter wavelength of line, by a 90 hybrid coupler, by lumped circuitry, or by proper selection of the operating points of the two multiturn loop antennas. The latter is illustrated in FIG. 18 where oneloop is operated somewhat below its resonance such that its input impedance is R +jX and the other just above resonance such that its input impedance is R jX. If, for example, X R and the loops are connected in parallel, the currents in the two loops will be equal in amplitude but differ in phase by 90 which is the desired condition. A series arrangement might also be used.

Thus, with no additional circuitry, the two loops mounted and operated in such array eliminate the usual minima associated with a single loop or dipole antenna. The physical position of the two loops should not be critical except for the orthogonal relationship. That is, the loops might be mounted in. separate cavities as v shownin FIG. 17, in the same cavity but physically separated as shown in FIG. 6, positioned one above the other as shown in FIG. 6B, orthogonally interwound as shown in FIG. 6C.

As described above relative to FIG. 9, a balanced tuning arrangement for the multitum loop antenna is effective and preferred in certain applications. However, it has now been determined that a much improved antenna is that with unbalanced turning.

With particular reference to FIGS. 1 and 26, an unbalanced feeding arrangement was used for the multiturn loop antenna l9mounted in a cavity 14. The center conductor of the coaxial feed is connected to one end of the loop antenna while the shield is connected to the cavity (ground plane) 10. The other end of the loop is also connected electrically to the cavity 10 such as through capacitor 27 in FIG. 26.. This type of feed eliminates the balun transformer and measurements can be made with standard equipment which has coaxial fittings; mechanical and electrical stability are also improved. Further, since the antenna appears longer by the image effect the resonance points occur at lower frequencies. In this way with the multiturn loop' antenna shown in FIG. 1, for example there arethree antiresonance points in the HF frequency band. This allows greater possibilities for matching the antenna to a specified impedance level.

It has been found desirable'to operate the multitum loop antenna at frequencies just below its natural antiresonances. Although it would appear that the antenna is limited to various frequency ranges in the regions of antiresonances, by tapping (shorting out) turns it is possible to shift the natural antiresonance points. This, in turn, shifts the points where capacitive matching is effective with an upward shift in antiresonances for 7 one, two, or three-turns tapped. Smaller shifts in operating points are obtained by tapping fractional turns.

Although the tapping technique is very effective, it requires an elaborate mechanical or diode-switching arrangement. It was found that the same effect can be produced by 'using a capacitive termination as 27 in FIG. 26. In this manner, the antenna electrical length is decreased and the antiresonance points shifted accordingly. It should be noted that inductive terminations produce the oppositeeffect of-increasing the electrical length of the antenna. However, since the use of inductors affords a loss inefficiency, only capacitive terminations are considered. Their effect is illustrated in FIGS. 14 and l9.

Therefore, with two variable capacitors, one in series with the feeding system and one acting as a termination as illustrated in F IG. 26, the antenna allows continuous coverage over a wide frequency range. The termination simply shifts the impedance curve'to the desired frequency while the series input capacitance nulls out the 20 resulting inductive reactance.


pacitance of.33 pf on the input side tunes out the 720 ohm inductive reactanc e. Measurements show that the impedance does become pure real. at about 6.66 MHz (actually 6.73 MHz) as shownin FIG. '20. The slighterror can be contributedto lumped element tolerances and measurement inaccuracies.

FIGS. 21 and 22 show the calculated input impedance as a function of 1/)\ through two antiresonances. The first antiresonance occurs at the frequency where l 0.58Awhile the second one occurs at the frequency where 1= l.74)t.

In order to verify the predicted characteristics, an experimental model was constructed. The measured inputjimpedance is also plotted as a function of l/A in FIGS.-2l and 22, and. the agreement with the predicted curves is clearly shown.

THEORETICAL AND ANALYTICALANALYSIS In analyzing the multitum loop antenna of the present invention, pulse basis functions are used. Each turn in the antenna is geometrically approximated by many short, straight segments. Since the segments are not all parallel to each other, the integral operator for this problem is morecomplicated. To be considered are both the axial and radial components of the electric FIG. 5,

vfield from each segment. That is, for the segment of Lawn) Qawfi E I 2 I I u) where L,,,, is defined by Eq. 9, I where N v 40 t l e mm] saw- The integration for E P 3 may be carried out in closed 15 form giving where r Q i (2a) These expressions are accurate ifp 0 or if r, and r are large in comparison with the quantity ap. For the segmentation problem of interesthere, these condi tions will hold true.

To calculatethe elements in the impedance matrix, we must'in effect calculate the tangential component of the electric field radiated by segment j'when the obser vation point is at the center of segment i. Details of this calculation are found below. The number of equations N is, of course, equal to. thetotal number of segments. Thus, the equation may be written as However, in this problem there exists a two-fold sym-' metry betw'een'the multitum loop antenna and its image.Thus the number of unknowns can be reduced by a factor of two and the i"."equatiojn may be written as i, i g I i n mm) i v l where I, 1

The quantity IEZ' represents the field due to the source. It is advantageous to use the magnetic frill current to represent the aperture where the coaxial cable joins the ground plane. Thus, an accurate modeling of anactual source aids in computing accurate impedance data that can be confirmed experimentally. For example, FIGS. 21 and 22 show both calculated and experimental values for the input impedance of a two and one-half turn version of .the multitum loop antenna. The agreement between theory and experiment for the reactance as well as the resistance is seen to be excellent. In the calculations, I26 segments were used.

more turns to the multitum loop antennawhile keeping the physical length of conductor constant and the same turn to turn spacing are threefold. That is to say, the

islowered somewhat, and the radiation resistance at nonantiresonant frequencies is reduced.

The implications of the gain and efficiency effects of any nearby structure are also important when considering the loop element. These same-loops have efficiencies of 50 percent or possibly more when placed on a metallic surface, such as an airframe, in the correct manner. To obtain these efficiencies, it is necessary that the loops be constructed with thick wire (i.e., on the order of at least l/ 100 of the wavelength of the operating frequency).

When the permittivity of the medium in which the multiturn loop isplaced is increased, the effect is to decrease the velocity of propagation of the currents on the loop. Hence, dielectric loading providesan effective means of increasing the electrical length of the antenna, and it can be employed in situations where size reduction is important. This also applies to ferrite load ing. I

lf it'were possible to surround the entire antenna with a total dielectric material space 6, the frequency shift would be given by ff! fo/ V ra where f,, is the original frequency and e, is the relative permittivity constant.

Experimental measurements on the dielectric loaded model of FIG. I tend to confirm the theory. With paraffin, e, 2.23, FIGS. 11 and 12 illustrate the shifts in resonant frequencies. At the higher frequencies the near fields of the multiturn loop are more tightly bound to the structure-and hence, in terms of electrical dimensions, the antenna is almost entirely surrounded by dielectric space, 6. Thus, the first antiresonance point is shifted down a smaller percentage than the second and third antiresonance points. i Efficiency measurements in the vicinity of the first antiresonance show that the 13 turn loop with an airfilled cavity has a peak efficiency of approximately 50 percent. FIG. 13 shows that with dielectric loading the antenna efficiency is reduced slightly and the peak value occurs in the vicinity of the shifted antiresonance frequency.

Although the multiturn loop antenna of the present inventon has numerous potential applications, it has defied a rigorous theoretical analysis. A prior analysis considered a balanced feeding arrangement and did not allow for cavity mounting Experimentally it has been demonstrated that the cavity walls have little effect in influencing impedance characteristics. Unbalanced feeding has the most pronounced in the sense that the antenna appears to be about twice as long as the physi-' cal length of conductordue'to image effect and antiresonance points occur at lower frequencies.

The experimentalcurves presented provide significant information for design. For instance, if the fact that the multiturn loop element has anti-resonance points at frequencies where the-antenna length (including image) corresponds approximately to ll/2, 3,u/2, 5/.L/2, etc., while allowing a loading factor of approximately 0.8 for the cavity effect and interaction between turns, such antennas may be designed for any specified frequency band. Simply using these two pieces of information, antennas have been successfully designed for operation at 40 and MHz. Once the antiresonance' point has beenplaced in the vicinity of the desired fre quency, the fine adjustment is obtained by a variable capacitor termination as described above.

The theoretical-numerical method is utilized for determining the current distribution and input impedance associated with the multiturn loop element. The total length of wire (including image) is divided into N (denoted FN in the computer'program) segments; the actual turns are approximated by M-sided polygons. If each wire segment is very short in comparison with the wavelength, it is reasonable to assume that the current density is uniformly distributed over the surface of each segment. That is, the true current distribution is approximated by a staircase function.

In a rigorous solution, the tangential electric field intensity vanishes everywhere on the surface of each perfectly conducting wire segment. If the wire radius is small, and the segments are short in terms of the wavelength, it is found that accurate'results can be obtained by forcing the total tangential electric field to vanish at just one point at the geometric center of each segment.

Since the complex value of the uniform current on each segment is unknown, it is necessary to generate a system of N equations in terms of N uniform currents. Generation of the necessary N equations can be accomplished by requiring that E=H+E where E is the total electric field at the surface or inside the wire conductor and represents the sum of the scattered field E and theincident field E.

Assuming thatthe electric field does not penetrate appreciably to the axis of the segment, E is taken to be zeroi Hence,

is the condition enforced at the center of each segment. E is the incident field fromthe primary source, while E is the scattered electric field caused by the induced current in the conductor. Thus, the problem is to determine the induced current value which satisfies Eq. (8).

The first step in solving the given problem is to accurately determine the value of the incident electric field, E, produced by an unbalanced coaxial feed. An unbalanced coaxial feed can be compactly represented by a magnetic frill current source. .The near and far-zone electric and magnetic fields from an annular ring of circumferentially directed magnetic current are calculated. For the near fields, numerical integration is used to obtain the electric vector potential, and then numerical differentiation methods are used to determine the near-zone field values. With suitable approximations,

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U.S. Classification343/708, 343/846, 343/745, 343/718, 343/787, 343/895, 343/744
International ClassificationH01Q7/00, H01Q19/09, H01Q1/28, H01Q19/00, H01Q1/27
Cooperative ClassificationH01Q19/09, H01Q7/00, H01Q1/286
European ClassificationH01Q19/09, H01Q7/00, H01Q1/28E