US 3078462 A Description (OCR text may contain errors) Feb. 19, 1963 J. HERMAN 3,078,462 ONE-TURN LOOP ANTENNA Filed July 18, 1958 2 Sheets-Sheet 2 INVENTOR JUL/U5 HERMAN W. a ma aafllw United States Patent-O 3,078,462 ONE-TURN LOOP ANTENNA Julius Herman, 5353 Chillum Place, Washington, D.C. Filed'July 18,1958, Ser. No. 749,570 2 Claims. (Cl. 343-744) (Granted under Title 35, U.S. Code (1952), sec. 266) Although loop antennas have been known for some time they have received relatively little attention in recent days. The chief reason for this is perhaps because'of the absence of a usable theoretical analysis of the opera tional characteristics of the loop antenna. Without such a usable theoretical analysis, prediction of the operation of the loop, or the'effects of various modifications thereof are well-nigh impossible. Any exploitation of the loop antenna thus never got oil the ground. As a result, the development of loop antennas today is not very much more advanced than it .was twenty years ago. I have developed a complete mathematical theory for the characteristics of a one-turn, thin-wire loop antenna. in air, andwith a finite: spherical core of material other than air. The theory holds for any size loop (as long frequency, and for any core material, including air, whose characteristics. are homogeneous and isotropic. Losses in the core are incorporated in the solution provided the volume current effect is negligible relative to the conduction current effect of the antenna, which is usually the case for practical cores. Expressions have been derived from this theory for. the input impedance, input power, radiated power, and. radiation efficiency; and a relatively simple, straight-- forward technique has; been developed for using these Ice 3,078,462 Patented Feb. 19, 1963 ,Still another object; is to provide a one-turn loop antenna having a controllable input impedance. An additional object of this invention is to provide a one-turn loop antenna having a spherical ferrite core whose radiation efiiciency is optimized. A further object is to provide practical embodiments in. accordance with the invention. 7 The above objects are accomplished by inserting distributed and/ or lumped impedances in theantennaloop, and choosing their values in accordance with the derived" equations to provide a desired set of antenna characas the loop can be considered a thin-wire loop) forany teristics. For the particular case of a uniformly distributed reactance inserted in the loop, it 'has'been discovered that the theory can be applied in an amazingly simple and straight-forward manner so that the solution of actual antenna problems becomes a practical reality. This simplicity results because the loop antenna has been found to act as a transmission line which supports an infinite number of modes. By inserting a properly chosen uni -formly distributed reactance in. the loop, a selected mode is made predominant, the antenna characteristics for the selected mode then becoming the characteristics of the antenna. This leads to a very great simplification in the design of loop antennas using the derived equations. The specific nature of the invention, as well as other objects, uses and advantages thereof, will clearly appear from the following description and from the accompanying drawing, in which: 7 g FIGURE 1 is a pictorial representation of the spherical I geometry employed in formulating the theory of the one: expressions to predict the eifect of various modifications on the loop. As a result, I have discovered that very important new results can be obtained by incorporating in the loop specially chosen lumped and/or distributed impedances. These results include control of the antenna input impedance, control of the antenna radiation pattern, optimization of the radiation efficiency of a ferrite core loop antenna, and tailoring of the antenna characteristics. As it is well known in the art such control of the characteristics of a loop antenna was heretofore not possible. This is because for a given loop geometry at a given frequency, the input impedance, radiation efficiency and radiation pattern were fixed and out of the control of the designer. Any attempted modifications of the loopwere made on a substantially trial and error basis. These trial and error modifications met with little success because the chances of achieving the combination of values which would produce a desired result were very small. It is the broad object of this invention, therefore, to provide means and methods for modifying a loop antenna so as to achieve predictable control over the antenna characteristics. This control is obtained where the loop antenna is used in air, or with a finite spherical core of material other than air. It is another object to provide a one-turn loop antenna having predictable control of its radiation pattern. A further object of this invention is to provide a oneturn loop antenna having a uniform radiation pattern in the plane of the loop, even where the loop perimeter approaches or exceeds a half wavelength. ' tribution I()' on the loop 10 was assumed to'co'nsist or turn thin-wire loop antenna. FIGURE. 2 is a circuit diagram of a circuit for adjusting the resistive input impedance of a one-turn loop antenna having a uniformly distributed reactance in the loopso as to match the output resistance of a generator. FIGURE 3 is a perspective view of a practical embodiment of the invention'wherein closely spaced lumped reaotances around the loop are used to simulate a uniformly distributed reactance. 1 FIGURE 4A is an enlarged view of one portion of the loop showing a lumped capacitance in detail. FIGURE 4B is a cross-sectional view taken along 4B--4B in FIG- URE 4A. FIGURE 5A is a front view of another practical e bodiment of the invention. FIGURE 5B is an end view of a portion of the embodiment of FIGURE 5A. To explain how to make and use the invention,- it is accordance with the invention will be presented, and actual examples of their use to solve practical antenna problems will be demonstrated. BASIC SOLUTION AND FORMULATION OF THEORY FOR A THIN-WIRE, ONE-TURN LOOP IN AIR In formulating the theory of the one-turn thin-wire loop antenna, the spherical geometry shown in FIGURE 1 was used. The wire loop 10 lies in the X--Y plane and the source of power of peak voltage V is applied at 0. The radius to the extreme edge of the loop 10 is c, the wire radius is a, and b is the radius from the origin 0 to the center of the wire. In deriving the mathematical solution, the current disa Fourier series in the variable 95 with unknown current coefficients a a and b as given below: unr) Mnk) Sin 0 eom It can be seen that the expressions are in the forms of v a product of functions of thethree coordinates 0, g5, and r.' The e dependenceis that of associated Legendre polynomials P with argument cos 0, the .r dependence is in the formof; spherical Hankelfunctions of thesecond kind H,, with argument fir, and the 5 dependence is in the form of trigonometric functions cos k andsin k b. The A F and G coefiicients are constants while n* is the complex intrinsic impedance of the medium. The electric field component is represented'by Bend the magnetic field component by H, the subscripts corresponding to the geometry of FIGURE 1. For any field component, the total field is the .sum of t-heccontribution from each mode o-fleach waye family. As in any antenna problem, the solution is now obtained by determining the coeflicients, a a and 'b in terms of known values, such as the applied voltage V,,, the frequency f, the internal impedance function 'Z of the Ioop,-a nd the loop geometry shown in FIGURE 1. In formulating the solution, the induced tangential elec tric field E; at the extreme outer edge of the loop 10 (r=c, 6:90") is firstcalculated from tthe'TM and TE lfll l lll H Tnk 2n(n-k) +k 1) 1 n-LX3 8) Theterms used in the above equations may be defi d as follows: n*=complex intrinsic impedance of the medium fi=complex phase constant of the medium v :fio\/( ..'-1'K..")(K.'iK.")' where: p =complex phase constant of free space f\/ o= M f=frequency u =permeability of free space e =dieleotric constant of free space A =wavelength in free space H fic)=spherical Hankel function of the second kind A J (flc) =spherical Bessel function with argument (pc) N 8c) =sp-herical Bessel function of the second kind with argument (5c) ii 'qsc) =derivative of H (fic) P )=associated Legendre polynomial with argument cos 6:0 A known boundary condition is now applied. This condition is that total tangential electric field E, at each point of the extreme outer edge of the loop (r=c, 6='90), is equal to the product of the current I() and the internal impedance function Z around the loop. Mathematical-1y this general relationship may be expressed as: where E; is the applied electric field resulting from the applied voltage V This relationship of Equation 9 is justified in the case of a thin wire where the electrical length B a 1. The terminal zone effect for the gap in the loop at which a voltage source V is applied may be eliminated by the use of the well-known mathematical concept of a slice or delta function generator so that the applied electric field Ef will exist only in the gap. This delta function generator is defined as an infinite electric field in a region of infinitesimal length having a line integral across the region which is finite and equal to the applied voltage V,,. When so defined the relation between the applied electric field Bf and the applied voltage V is as follows: where 6( is the translated delta function, 5,, is the point in the loop 10 at which the voltage source V is applied, and V is the applied voltage source V expressed as the sum of an infinite number of continuous functions. The property of the translated delta function 6( is such that: 6 Since in FIGURE 1 the voltage source V is applied at =0', Equation 10 becomes: Now substituting Equation 2 for E and Equation 12 for E4 in Equation 9, the following equation is obtained: +h2:;. k[ak cos lap l-b sin M] (1 Equation 1 for the current distribution I() is repeated below: i Using Equations 1 and I4 and well known mathematical procedures, such as the use of orthogonality relations. the coetficients a a and b can now be calculated in terms of known values for any internal impedance function Z around the loop. To obtain Equation 14 in a more practical form the impedance function of Z will now be expressed as the sum of a continuous distributed impedance function. Z around the loop and a lumped impedance function Z as follows: , n.()-lr.() The impedance function Z is continuous and can readily be handled in the solution of Equation 14. This function Z represents the presence of a continuous impedance function around the loop. The function Z on the other hand, represents the presence of lumped impedances in the loop so that periodic impedance discontinuities exist. To conveniently handle these discontinuities in' Equation 14 they will be approximated in a manner similar to that used to represent the driving source; that is, as impulse sinks utilizing the translated delta function as shown below: I() is the current at the lumped impedance Zm. The In index of summation refers to the number of inserted impedances. Substituting Equation 15, l6,v and 1 into Equation 14 gives Using orthogonality relations Equation 18 can now read- 11y be solved to give the values of a o and b in terms of known values for any distributed impedance function Z and/ or lumped impedance Z inserted in the loop. Each V() can be determined by substituting for each I() terms of the known impedances (perfect conductor impedances) and applied voltage. Once the values of a a and b are known, the input admittance =7 (and the -iput impedance Z which is the reciprocal thereof) may then be obtained as follows: Using the complex Poynting vector method, the radiated power P then follows as; (assuming free space for air) EXTENSION OF THEORY TO THE CASE OF -ATHIN-WIRE, ONE-TURN LOOP WITH A SPHER- ICAL CORE OF MATERIAL OTHER THAN AIR Once the solution for the loop in air is determined, the solution for the multimedium'problem follows from well known mathematical procedures which can be found in standard texts on electromagnetic theory. The spherical core is assumed'to be a material other than air where the outer edge of the loop coincides with the surface of the core;:that is, the radius r of the core isequal to'the outer radius of the loop shown in FIGURE 1. Upon solution it is found that the only equations previously presented which are aifected and modified areEquation 3 for K Equation 5 for K Equation 24 for PORAD and Equation 25 for P These modified equations are presented below. The superscript 0 will be used to indicate that these equations are for the case of a spherical core other than air. The modifying factors M M M are defined by the following equations: In the above'solution'of the multimedium problem-it has been assumed that the characteristics of the'core medium are homogeneous and isotropic. Losses in the core are incorporated in the solution as indicated by the appearance of the complex intrinsic impedance in the equations. In this connection itis assumed that the volume currentin the core is negligible relative to the conduction current efiect of the antenna. For practical 'core mediums (ferrite for example), the above'assumptions are valid. PHYSICAL INTERPRETATION OF THE BASIC THEORY From Equations 19 through 25 it can be'seen that the input admittance Y the input power P the radiated power P and the radiation efiiciency E are expressed as the sum of individual contributions from each component of the current I(,). The loop, therefore, may be considered to act as a transmission line which supports an infinite number of k modes of transmission with each mode having its own set of characteristics and each presenting to the driving source an impedance in parallel with those of all other modes. Furthermore, the .Equations.23, 25 and 29- 3O for radiated power indicate that the radiation may be considered as being comprised of an infinite number of n modes of spherical wavefront radiation emanating from the center of the loop. These k and n modes are interdependent as shown by the summation limits in the radiated power equations. 9 USING THE DERIVED THEORY FOR LOOP ANTENNA DESIGN some of the equations appear quite fearsome at first glance, it has been found that numerical answers can be obtained therefrom using standard mathematical procedures, even though the calculations are somewhat laborious. By programming the more tedious calculations on a computer, the time required can be considerably reduced. An analysis of the terms involved in the pertinent equations shows that the major difficulty is introduced by the spherical Bessel i and Hankel H functions of complex argument such as appear in Equations 3-5. Such functions have never been tabulated because of the obvious infinity of values for the argument. However, they can be calculated by two well known methods: the first is the use of recursion formulas, and the second is the expansion of each order in polynomial form. The calculation of K; which is the sum of three summations in Equation 5 might appear particularly difiicult, but is readily calculable by assigning values to k, beginning with zero, and summing each component of K over n for each k. The order of n is of course taken sufiiciently high to yield a close approximation to the limit. The same criterion also applies to k for the 1 1: summation. It can be seen, therefore, that although the necessary calculations are somewhat laborious, they are readily performed and may thus be employed for loop antenna design. Using the above-described calculation procedures, I have calculated the effect of various combinations and types'of distributed and lumped impedances inserted in the loop. These calculations showed that very considerable changes can be produced in the characteristics of the antenna, even by the insertion of a single lumped impedance. It is possible, therefore, by theoretically calculating the effects of various combinations and types of inserted impedances, and constructing a one-turn loop antenna in accordance therewith, to arrive at a set of antenna characteristics which will be most favorable for a given application. The previous rigidity of performance which was heretofore characteristic of the loop antenna is thus at an end. While the above-described equations permit the characteristics of a one-turn loop antenna to be predicted for any type and combination of distributed and lumped impedances, the reverse procedure of determining the inserted impedance combination which will give a desired antenna characteristic is not readily calculable. In some instances this may pose a serious practical problem in the use of the derived equations where a particular antenna characteristic is desired. 1 have discovered that this difficulty is overcome to a very considerable extent by using a uniformly distributed reactance as the impedance inserted in the loop. The use of this uniformly distributed reactance leads to a tremendous simplification in the use of the derived equations, whereby the inserted reactance which will achieve a desired antenna characteristic can be determined directly from the derived equations. This will be brought out by the following discussion. APPLICATION OF THE THEORY TO THE PARTIC- ULAR CASE OF UNIFORMLY DISTRIBUTED REACTANCE INSERTED IN THE LOOP (1) Perfect Conductor Values To apply the theory to the case of uniformly distributed reactance inserted in the loop, Equation 14 will first be solved for the case where the loop is assumed to be a perfect conductor, that is Z ()==0. Using orthogonality relations, the solution of Equation 14 for a a and b The additional subscript s is used to indicate that the values are for the perfect conductor case, and will be so used hereafter. The equations for Y,,, P and E then .are determined by substituting Equations 34, 35, and 36, in the previously derived Equations 15-26 or 27-30 for a core other than air. (2) Derivation of the Equations in Terms of Perfect Conductor Values for the Case of Uniformly Distributed Reactance Inserted in the Loop InEquation 14 the internal impedance function Z is now assumed to be a uniformly distributed reactance X per unit length around the loop. The total internal distributed reactance X may then be expressed as: X =21rX (37) The insertion of the total distributed reactance X in the loop carries with it the practical fact that a loss component is associated therewith. The total uniformly distributed impedance inserted will thus be represented as: where R is the total loss component associated with X The Q is then Again solving Equation 14 using orthogonality relations, the equations for non, a and hi for the uniformly distributed reactance case then become: V0 Z05 v "ZT+ZO. Zo.+zT] 9 The equations for Y Pm P and E 'then follows as: ems-res 11' where: Z RAD RAD Zbs+ZT ks 2 nAD nAD (48) RAD oD in kD kt in Inthe above equations the subscriptD is used to indicate that these values are for the case of a uniformly distribut ed reactance inserted in the loop. After performing; a considerable number of calcula-- tions for the caseof a uniformly" distributed reactance X inserted in the loop, I'discovered that when the reactanceX- is chosen so that a particular modeimpedance- [Z orI Z inEquations 43 and-44]. isapproximately a pure'resistance, the characteristics .for that mode. then become'predominant, the effects of all. theother' modes becoming negligible. This has been found to occurb'ecause the reactive portions of the mode impedances are quite-large relative tothe resistivep'ortions, and-in addition, are widely. different-from one'another. It: is: thismode predominance effect, which occurs when a-runi formly distributed reactance isinserted .in the loop; which brings about a great simplification in the use of the derived equations. As can be seen-frornrEquations 43 and-44,,'a. selected mode impedance may be made predominant merely by making X equal and opposite to the reactive component of the selected mode impedance. For a given antenna the perfect conductor values Z and Z are known so thatv the. values of: X which will. make. a selected mode predominantismeadily. ascertainable; Thus; from: Equations 38-50 the. antenna characteristics: that. would. re-v sult if each mode were made predominant are also readily ascertainable.. It is apparent, therefore, that it is now possible to choose the mode that will have the most favorable characteristics-for a desired application. Since these characteristics vary-Widely for the different modes of a given loop, awidenumber of possible characteristics are availablefor selection by the designer. And, since the;characteristics of each mode are a function of known values, such as 11*, )8, K and K all of which are known. constants, these values maybe chosen so that a selected. mode will have practically any desired set of character'- istics. Some solutions to practical loop antenna problems which are possible by insertinga properly chosen uniformly distributed reactance in the loop will now be presented. SOLVING PRACTICAL LOOP ANTENNA PRO-B- EEMS BY INSERTING A PROPERLY CHOSEN UNIFORMLY DISTRIBUTED REACTANCE IN THE LOOP (1) Input Impedance Control One of the serious problems attendant with antennas in general is the inability to adjust or control the input impedancepresented by. the antenna. It is usually necessary, therefore, to employ impedance matching networks to provide proper matching. Using a uniformly distributed reactance X inserted in a one-turn loop antenna, this input impedance problem is now solvable. 'For a given antenna, the mode which gives the most. favorable input impedance may be. made predominant. Further, the constants n*, ,8, K and K may be ad= justed so that the selected mode will have the desired impedance. It is also possible to choose the total distributed re 12 actance X so that the'selected mode provides a net reactance instead of a pure resistance Where such is desired for matching purposes. This is possible because in most casesthe reactances of the non-selected modes will be sufliciently largerthan the desired net reactance so as to stillzmaketheir, effectnegligible relative to the selected mode. (2) Radiation Pattern Control Where-the perimeter of; a loop antenna approaches or exceedsxa half wavelength, it hasbeenheretoforedifiicult to obtain a radiation pattern which is uniform (circular) in'thesplaneof the; loop, and at the; same time, horizontally polarized;(that is,- .E, is uniform in the-plane of the loop). This has been accomplished in the prior art only by means of a variety ofcomplex arrangements such as thewellknown Alford loop. (See Electronic and Radio Engineering, 4th ed., by Terman, p. 908); This difficult problem is now readily solvable by means of the present invention. As can be seen from the TB andTM families, the E and H.fields are proportional to thecurrent coefiicients, a a and b in Equation lfor the current 1(4)). For the condition where thecurrent is uniform aroundtheloop (that is, I ().=a a uniform'E field'will be obtained in the-p direction (that is, in'the plane of the loop). In the 0 direction uniform current I() will resultin an 13 rfi'eld having approximately a figure eight pattern. The field pattern for the E; field will'thus have the form of a doughnut with its axis coinciding with the loop axis. It is evident that the condition for uniform current I()=a is readily accomplished for the case ofa-uniformly distributed reactance inserted in the loop. The current'coefiicient b is already zero [Equation 41], and. the coefficients a can bemade negligible by making the: zero mode a predominant; that is, by making X equal and'oppositeto the: reactive component of Z in Equation 43.. Since-thetheoryholds for any'sizeloop, this method of: obtaininguniform current around theloopalsoholds, for anysizev loop, including the case. where they loop, perimeter approachesor exceeds a half wavelength. At this.timeitzisinteresting tonotethe type of radia- :tionv pattern, obtained for the. case where the a current mode.(k= 1) ismade predominantby making, equal and opposite to thereactive component of Z see Equation 44]. For thiscondition the E field is uniform. in the 0 direction and approximately a figure eightinv the. direction;v The E, patternrwill thenhave the form. of adoughnut with its axis. perpendicular to the loop:axis. and: lying; symmetrically along the. 0 to lineinthe: plane. of thezloop. For theE field, thepatternin thee- (3) Optimization of the Radiation Efiiciency of a Ferrite Core Loop Antenna Another practical utilization: in accordance with this invention is the optimization of the. radiation efficiencyof a loop antenna having a ferrite spherical core. An analysis of" the derived equations shows that the radiation etficiency obtained for the condition of uniform current around the loop, I (-)=a not only provides a uniform radiation, pattern in the plane ofthe loop but also provides maximum radiation efficiency for. the case of a spherical core. Thus, to maximize the radiation ef- 13 14 ficiency of a ferrite loop antenna the zero mode is made B predominant. Since low radiation efficiency has been a serious drawback to the use of ferrite cores, this metho Po Pi iu-l of optimizing radiation efficiency is of considerable importance- 15.520 W 4.877 V 7.33s va 1.30s V0 0. 573 v0 I APPLICATION OF THE EQUATIONS FOR A ONE- TURN LOOP HAVING A UNIFORMLY DISTRIB- O UTED REACTANCE TO A SPECIFIC NUMERICAL HAMPLE P351)" RAD-l nanq RAD-r BAD-H To illustrate how the uniformly distributed reactance loop antenna can be advantageously applied in antenna 1. 522 W 1. 303 V0 0158 Va 0.00003 v0 design, a specific numerical example will now be given. It is to be understood that this will merely be a single D example of many possibilities, and in no way is intended to limit the scope of the invention. For this example it 13,35 will be assumed that the antenna comprises a one-turn erc'n emirt percent percent percent loop made of silver-coated bronze wire having a wire radius a=0.02l inch, an outer loop radius c=l.6485 in- 9.81 27. 95 2.15 i ches, and a Stackpole 2285 ferrite core with a core radius of c=l.6485 inches. For the above-identified ferrite medium the values at It will now be 'shown how the k=0 mode is made 170 megacy-cles of the parameters K K K and K predominant and the results obtained thereby. As de- (which are the previously defined non-dimensional real scribed previously, the zero mode predominance condiand imaginary components of the permeability and dielection achieves a desirable uniform radiation pattern. tric constant, respectively) have been obtained from actual And, where the losses consumed in the loss component experimental data by well known methods. The values of R associated with the inserted uniformly distributed rethe complex intrinsic impedance 77* and the complex phase actance X are not too large, zero mode predominance constant 190 may then be calculated as shown in Equaalso achieves an optimized radiation efficiency E tions 8(a) and 8(b). These values of K K K To achieve zero mode predominance the inserted unin? and 130 at 170 megacycles are given in Table I below: formly distributed reactance X is chosen so that the zero TABLE I Frequency K., Kn" K. K. n" B6 170 megacycles... 10.75 0.227 9.5 0.077 127. aim-0. 4222 1. 5068/(). 830 Using the above table, the perfect conductor values Z mode impedance Z is a pure resistance. From Table P P and E are calculated. Since the loop IIIA, Z '=2.384+j494.4 so that to make Z a pure necessarily has a self internal impedance due to skin efresistance, Equation 43 shows that X- should be a capacifect and ohmic resistance, this should be taken into ac- 45 tive reactance equal to 494.4 ohms. It can be seen count. The self internal impedance is of course uniformfrom Table III and Equations 38-50 that for this value 1y distributed and may be designated as Z =R +jX of X inserted the effects of all other modes are negligible, Calculating Z by well known means and substituting the the zero mode thus being predominant. The uniform calculated values therefor in Equations 38-41, initial radiation pattern is now the radiation patternof the anloop values of Z' P P and E are ob- 0 tenna and the resulting zero mode characteristics with tained. These primed values are now used in Equations X inserted are now the characteristics of the loop an- 38-41 instead of the perfect conductor values. The cal tenna. Assuming a reasonable value of Q =400 for the culated initial values at 170 megacycles are shown in inserted capacitive reactance X =494.4 ohms, the as- Table II below. sociated dissipative component R is substantially 1.2 'ohms. Using these values of R and X in Equations TABLE H 38-50, the following zero mode values (which are now 7,. PL P those for the antenna) are obtained as shown in Table 611321; micrd wri tts mici h ii' at ts perce nt IV below: l154+j6003 15. 520 V0 1. 522 V. 9.81 TABLE Iv ZOD= in-D, POin D=Piu-d, PORAD-D=PRADD, EOD=ED, ohms microwatts microwatts percent In Table II and hereinafter, it is to be understood that the source voltage V is in peak volts. 6 3.585-l-j0 0.1395 VD 0.0260 V0 18.6 The mode values for Z P P' and E 5 up to k=3 are given in Table III, A, B, C, and D below. From Tables I and IV it can be seen that the radiation 15 efiiciency has increased from 9.81% to 18.6%. Also, the input impedance has been radically changed from ll54+j6003 ohms to the very much smaller value of 3.584 ohms. Itis also possible to provide any other higher antenna resistive input impedance than 3.584 ohms by means of the circuit shown in FIG. 2. The numeral 11 represents the one-turn loop antenna with the total uniformly distributed capacitance X inserted in the loop. A variable-capacitor 11 is inparallel with the one-turn loop antenna 12, and the parallel combination thereof is fed by a generator 19 having an output resistance R. To match the generator output resistance-R, the value of the total inserted distributed capacitance is chosen so that the input impedance of the one-turn loop antenna 12 is the resistance R in parallel with an inductance. The variable capacitance 11 is then adjusted to tuneout this inductance, the input impedance presented by the parallel combination of the one-turnloop antenna 12 and the variable capacitance 11 then being equal to the generator output resistance R. As explained previously, the zero mode net reactance required for matching purposes will ordinarily be sufiiciently-small ascompared to the reactances of the other modes so as not to efiect the zero mode predominance-condition. For example, if an input resistance of say 100 ohms were desired instead of 3.584 ohms, this value can be obtained by inserting in the loop a capacitive reactance of X =479.2 ohms instead of'494.4 ohms. The input impedance Z then becomes 3.584+j15.2 ohms which is equivalent to aresistance of 100 ohms in parallel with an inductance of 15.7 ohms. This parallel inductance is now tuned out by the variable capacitance 11 so that the remaining antenna input impedance is a resistive value of 100 ohms as desired. From Table IIIA it will be seen that the zero mode is still predominant for this value of X =479.2 ohms inserted in the loop. The above-described illustrative example thus shows how a one-turn loop antenna can be tailored to provide a uniform radiation pattern in the plane of the loop, .optimum radiation efllciency, and practically any de sired resistive input impedance merely by properly choosing the total uniform-1y distributedreactance X inserted inthe loop. PRACTICAL EMBODIMENTS OF .A LOOP AN- TENNA HAVING UNIFORMLY DISTRIBUTED REACT-ANCE INSERTED IN THE LOOP Practical embodiments of a one-turn loop antenna having uniformly distributed reactanoe in the loop have been built and tested. In building and testing these embodiments it has .been determined that various practical assumptions arepossible without upsetting the validity of thederived theory. First, it has been determined that a uniformly distributed reactance may be approximated by using closely spaced lumped reactances around the loop as is sometimes done in building an'artificial transmission line. Secondly, it has been found that the radius of the ferrite core-used is not critical, and the calculated results will not be significantly altered as long as the wire loop is in close proximity to the core. Resting the loop on the surface of the core has been found to be entirely adequate. Thirdly, as regards the value of the self internal impedance of the loop Z '=jX '+R to be used in Equations 38-41 to obtain the initial values of the mode characteristics with no inserted uniformly distributed impedance, it has been found that the value of X;- is maintained as long as the perimeter at each cross section of the actual loop is approximately equal to that of the round wire. That is, the X value is the same as that for a round wire as long as the perimeter at each cross section is approximately equal to 2711:, where a is the radius of the round wire. Each reactanoe of any set of lumped reactances inserted in the loop to approximate a uniformly distributed impedance, therefore, should have a perimeter substantially equal to 21m. The value of X in the equations Will then bethe sameas for a. round wire having the same perimeter, and th-e reactance value of the inserted impedance (excluding leads) will then serve as the inserted reactance. The value of the ohmic resistance R canreadily be'calculated m-easured, or estimated by'well kn'own means. The value of R ordinarily has only a minor cfiect on the resultant initial values. FIGURE 3 shows anumber of closely lumped capaci'tances 20 inserted in a one-turn loop 15 to approximate a uniformly distributed reactance. FIG. 4A is an enlarged view of one portion of the loop 15 showing one capacitor in detail, and FIG. 4B is a cross-sectional view along 4B-4B in FIG. 4A. To achieve approximately the value of X =494.4 ohms total internal distributed reactance at megacycles required to make the zero mode predominant in the numerical example, twenty-five capacitors each having a value of 50 micromicro farads at '170 megacycles are employed as shown. As in the-numerical example, the radius c to the extreme edge of the loop is chosen to be approximately 1.6485 inches, the perimeter 2w+2t is chosen to be approximately equal to and the core 30 is made of Stackpole 2285 ferrite with a core radius such that the loop '15 rests snugly on the core 30. The applied voltage V is applied to terminals 70 forming about a A inch gap in'th'e loop 15. As shown in FIGS. 4A and 4B, the plates 45 of the capacitor 20 are connected to the connecting wires 35 by suitable means such as solder joints 4.0. The dielectric of the capacitor is represented by 50. The value of the capacitor 20 used in determining the distributed reactance is the capacitance of each unit 20 only, without the connecting leads 35. With the embodiment of-FIG. 3 constructed as .described above, experimental data shows that the zero mode is predominant and the antenna characteristics obtained are remarkably close to those predicted by the numerical example. FIGS. 5A and 5B show another embodiment of the invention for achieving a uniformly distributed capacitance in the loop. A dielectric loop 55 of rectangular cross section is snugly fit around a spherical core 30. Electrodes 61B are formed on the dielectric loop 55 so as to be symmetrically and alternatively located around the inner and outer peripheries of the loop, thereby providing in elfect a uniformly distributed series capacitance. As was done for the embodiment of FIG. 3, the radius '0 to the extreme outer edge, the perimeter v2w-I-2t, and the ferrite core 30' may be chosen to provide the initial values shown in Tables III and IV, and the value of the inserted uniformly distributed capacitances may likewise be adjusted to provide zero mode predominance. The size and spacing of the-electrodes 60 in cooperation with the dielectric loop 55 which are necessary to provide a given uniformly distributed reactance may readily be determined by well known means. The spherical ferrite core employed in the above two embodiments was constructed from seven ferrite disks about three and a half inches in diameter and about onehalf inch thick. These seven disks were ground to optical flatness and then mechanically clamped to form a cylinder about three and a half inches long. The cylinder was immersed in heated melted parafiin and after cooling the excess paraflin was wiped off. The cylinder was then gripped with a U clamp on its end surfaces and ground to a sphere using a diamond wheel grinder. It will be apparent that the embodiments shown are only exemplary and that various modifications can be made in construction and arrangement 'within the scope of the invention as defined in the appended claims. From the desirable results obtained with the one-turn 17 loop antenna by inserting distributed and/ or lumped impedances in the loop, it is felt that the proper use of inserted impedances in other types of antenna configurations will also lead to new and desirable results. I claim as my invention: 1. In combination with a single-turn loop antenna, means for supplying current at a predetermined frequency to said loop and impedance means substantially uniformly distributed around, and forming a part of, said loop for causing a predetermined k current distribution mode to predominate at said frequency independently of the electrical length of said loop, where k is any integer equal to or greater than one, wherein said impedance means comprises a reactance X inserted in said loop having a value equal and opposite to the reactive portion of Z the mode impedance, where 1L Z equals K Where V is the voltage impressed at the input terminals of said antenna and I(), the loop current in the direction, =a +2(a cos k+b sin k) where a a and b are the co-efficients of a Fourier series representation of the current wave I(), and is the angle between two lines lying in the plane of the antenna and passing through the geometrical center of said loop, where the first of said lines also passes through the point on said antenna where the input voltage is applied and the second of said lines passes through any other arbitrary point on said loop. 2. The invention in accordance with claim 1, wherein said reactance X is approximately equal and opposite to twice the reactive portion of the initial first mode impedance Z of said loop, where Z represents the first mode impedance defined in claim 1 with a correction for the fact that said loop antenna is not a perfect conductor, thereby making the k=l mode predominant. References Cited in the file of this patent UNITED STATES PATENTS 977,462 Matthews Dec. 6, 1910 2,166,750 Carter July 18, 1939 2,266,262 Polydorofi Dec. 16, 1941 2,311,872 Rote Feb. 23, 1943 2,316,623 Roberts Apr. 13, 1943 2,501,430 Alford Mar. 21, 1950 2,642,529 Frankel June 16, 1953 2,881,429 Radclifie Apr. 7, 1959 2,887,682 Charman et a1 May 19, 1959 OTHER REFERENCES Book Radio Engineering by Terman, 3rd edition, McGraw-Hill Book Co., page 691, TK6550T4. Kraus: Antennas, McGraw-Hill Book Co. New York, 1950, pages 4-8 referred to. Leeds: Abstract 145, 353: 665, 0.6. 1316, Dec. 23, 1952. Patent Citations
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