COMMUNICATIONS 697

TABLE I1

C, CONSTWTS (m\’.s) ~~

>\‘ DKO DK I KO K1

9 14.0+j78.8 12.3+j75.S 10.7+j72.3 19 12.8fj76.5 12.l+j75.2 11.2+j73.1

11.4+j73.7 11.6fj74.2

29 39

12.3+j75.7 12.l+j75.2 11.5+j73.8 11.8+j74.5 12.3+j75.5

49 12.1+j75.2 11.6+j74.1 11.8+j74.6

12.2+j75.4 12.2+j75.3 11.7+j74.3 11.9fj74.7 59 12.2+j75.3 12.2+j75.3 11.8+j74.5 12.0+j74.R 79 12.2+j75.3 99 12.2+j75.3 12.2+i75.4

32.2+/75.4

Table I also exhibits that the DKO and KO low-order solutions are comparable in ac- curacy. The C , constants as well compare very favorably.

ACKNOWLEDGhlEXT

The witers are indebted to Dr. C. W . Harrison, Jr., for many enlightening discus- sions, and to Barbara Ford for typing the manuscript.

E. A. ARONSON C. D. TAYLOR Sandia Corp.

Albuquerque, N. Mex.

REFERENCES [l] R. F. Hanington, “,Matrix methods for field prob-

lems,” Proc. IEEE, vol. 55, pp. 13G149, February

[2] R. W. P. King, “Linear arrays: Currents, imped- 1967.

an-, and fields, I,” IRE Trans. Antefinus and Propagation, vol. AP-7, pp. S4.M-S457, December

131 C . W . Harrison, Jr., C. D. Taylor, E. E. O’Donnell, 1959.

and E. A. Aronson. “On digital computer solutions of Fredholm integal equations of the first and see- ond kind occurring in antenna theory” (submitted for publication in Radio Science).

Author’s Repb‘ The observations by Aronson and Taylor

agree with those obtained from our studies. I should like to present an additional pro- cedure which we have found to give results comparable to the K1 technique above, but which uses the simplicity of the KO formula- tion. This involves flat-zoning (called a step approximation in our work) and constraining every other current coefficient to be the average of its two adjacent coefficients. The matrix to be inverted is thus reduced to one- half its original size, and the resultant ac- curacy is comparable to that of the K1 method. This constraining technique is par- ticularly convenient when the antenna is

1 Manuscript received April 12. 1967

curved, in which case the K1 and K2 integra- tions becomes quite difficult. Further discus- sion of such questions can be found in my forthcoming book, Field Computation by Momerzt Methods, to be published by Mac- millan in November 1967.

ROGER F. HARRINGTON Dept. of Elec. Engrg.

Syracuse University Syracuse, N. Y.

A Parasitic End-Fire Array of Circular Loop Elements

This array is related to both the Yagi-Uda and Cigar antenna [l]. Its elements are circu- lar-conducting wire loops approximately one wavelength in circumference. Its operation is best described by first considering a linearly polarized version. The extension to circular polarization will be shown shortly.

Consider first the current distribution on the driven element. Adachi and Mushiake [2], [3] have used an integral equation and its solution to determine the current distribu- tion. For a loop of approximately one wave- length in circumference, the current has lossy transmission-line character and is approxi- mately of the form of a hyperbolic cosine with complex argument [2]-[4]. The current distribution is still very nearly a sinusoid, with radiation accounting for the equivalent transmission-line losses. The current distri- bution for a loop 1X in circumference is shown in Fig. 1. It is best visualized when the loop is considered to have been formed by deforming a shorted one-half wavelength parallel-wire lossy transmission line into a circle. The loop will have a far-field radiation pattern which is maximum broadside to the

1967. Manuscript received April 6, 1967: revised April 21,

loop (Le., along the axis of loop). The polari- zation will be linear and parallel to an imagi- nary line connecting the two current nodes. It is interesting to note that the (broadside) directivity of a one-wavelength loop is ap- proximately 4 dB above isotropic, i.e., about 1.8 dB above a one-half wavelength dipole [3]. This differential of 1.8 dB (in favor of the loop array) also appears when a parasitic array of loops is compared to a Yagi-Uda array. This has been experimentally verified by comparing the E and H plane (3 dB) beam nidths of the two arrays with the array length of the loop-array made equal to that of the Yagi-Uda array. Fig. 2 shows this comparison between a circular loop and Yagi-Uda arrays. The plotted points indicate gain versus array length for the two types of arrays, where

30 000 EerrHerr

G = -,

with EBW= E-plane half-power beam width in

HEW= H-plane half-power beam width degrees

in degrees.

These measurements were performed at 440 MHz.

FEED

Fig. 1. Current distribution on a circular loop one wavelength in circumference.

Fig. 2. Experimental comparison between circular loop and Yapi-Uda arrays.

As with the Yagi-Uda array, the shape factor of the elements determines theexact dimensions. By empirically fitting the data given by Adachi and Mushiake [3] it can be shown that a practical circumference correc- tion factor F is given by

F = l + - + - 0.4 3.0 Iz 0 2

IEEE TRANSACTTONS ON ANTENNAS AND PROPAGATION, SEPTEMBER 1967 698

where

with C=loop circumference, and 6=wire di- ameter. The resonant length of the driven ele- ment is then given by F A , where h is the free- space wavelength.

A parasitic array is now formed bymak- ing the reflector element about 3 percent longer than the resonant length of the driven element and the directors about 3 percent shorter. Element spacings are typically those used in a Yagi-Uda array. The bandwidth of these arrays is similar to that of the Yagi-Uda array. Experimentally it can be shown that combinations of linear and loop elements produce workable parasitic arrays with gains somewhere between the Yagi-Uda and loop arrays.

Circular polarization can be obtained in a variety of ways. The most direct way is to use a crossed dipole-fed element. All other ele- ments are left unaltered since they have the necessary circular symmetry.

Another approach would be to use square loops by placing them in the configuration shown in Fig. 3. The indicated arrangement of square loops is chosen so as to reduce mutual coupling between the two feed loops. It should be noted here that experimentally the square and circular loops (of 1 X circum- ference) have essentially the same radiation and impedance properties.

FEED I

Fig. 3. Crossed-loop feed for circular polarization.

A third approach is to feed a single loop at two different appropriate points on the loop. Experimental work would be required to determine the feasibility of this scheme and ascertain whether mutual coupling effects would be deleterious in the crossed-loop scheme. Other feed mechanisms are possible and should be further investigated.

The main advantage of this array is its slightly higher gain over that of a Yagi-Uda array. This allows the loop array to be some- what shorter than the Yagi-Uda array, an irn- portant consideration in the HF range. Con- sideratio2 of the experimental data given in Fig. 2 shows that the Yagi-Uda array must be approximately 1.8 times longer than the loop array to obtain the same gain.

Its other characteristics and properties are very similar to those of the Yagi-Uda parasitic array.

ACKNOWLEDGMENT The iirst known application of a linearly

polarized parasitic arrangement of loops ap- pears to have been made in the summer of 1942 by C. C. Moore at Radio Station HCJB in Quito, Ecuador, on the 25 meter band. An existing Yagi array had suffered from an extreme corona problem whenever full power was applied to the antenna located at the 10 000 foot altitude of Quito.

The parasitic loop array consisted of two elements, a driven element and a parasitic reflector, and was reported to have completely eliminated the corona problem. Other South American commercial shortwave stations have used similar arrays; for example, Radio Station TGNA, operating on the 49 meter band in the 5OOO foot altitude of Guatemala City, Guatemala.

This array appears to be very popular in amateur radio circles and is called a “cubical quad” or “quad” antenna [ 5 ] , [6]. The term “quad” is supposedly due to the square-loop and box-like construction which is normally used.

JAMES E. LINDSAY, JR. Martin Company

Denver, Colo. 80201

REFERENCE

[ I ] Jasik, Alttenrla Engineering Hmdbook. New York:

[2] S. Adachi and Y. bluchiake, “Theoretical formula- McCraw-Hill, 1961, pp. 16-25.

tion for large circular loop antennas by integral equation method,” Repts. of the Research Institute of Elec. Commun., Tohoku University, Sendai, Ja- pan, Sci. Rept. Ritu, B (Elec. Commun.), vol. 9, no.

I31 -. “Studies of large circular loop antennas;’ 1, 1957.

Tohoku University. Sci. Rept. Ritu, B (Elec. Com- Repts. of the Research Institute of Elec. Commun.,

141 J. E. Lindsay, “A circular loop antenna witb non- mun)., vol. 9, September 1952.

uniform current distribution;’ IRE Tram. Antennas and Propagation (Commwlicatiom), vol. Ap-8, pp.

[5J W. I. Orr, Quod Antennas. Wilton, Conn.: Radio 4 3 9 4 1 , July 1960.

14 L. Bergen, “A multi-element quad antenna;’ QST, Publications.

%fay 1963.

Graphical Aids for Polarization Problems

With reference to the recent discussionl** on graphical aids for polarization problems, a third way of mapping the PoincarC sphere may be found useful. This method consists of projecting the sphere from the point L1 onto the plane tangent at Lt in Knitters Fig. 1.’ Since this is a stereographic projec- tion, it will retain the advantage of conformal mapping, and Knitters results may be ap-

Manuscript r e i v e d April 6. 1967.

ical aid in determining the polarization of an antenna 1 G. H. Knittel, “The polarization sphere as a graph-

by amplitude measurements only,” IEEE Trans. AN- remas andPropagation, vol. AP-15. pp. 217-221, March 1967.

9 J. Ruze and G. H. Knittel. Discussion on Knittel’s paper,’ IEEE Trans. Antennas mld Propagation, vol. AP-15, p. 221, March 1967.

plied with obvious and unsubstantial modi- fications. The additional advantage is that this projection plane has a physical meaning. It is the complex plane of the polarization factor p , which gives the relative amplitudes of, and the phase shift between, two orthog- onal linear polarizations (e.g., horizontal and vertical) in the directions of the unit vectors e- and e+. Let a plane wave be given

E = E-e- f E+e+, (1)

bY

where the factor exp (iwt-k‘r) has been suppressed and E-, E+ are generally complex. Setting

E+ E-

p =-,

then (1) can be written as E = E-(e- + pe+). (3)

It is evident from (2) that IpI gives the ratio of the amplitudes and arg p gives the phase shift between the two components, and that any poh-ization can be uniquely de- scribed by p ; e.g., linear: Im p=O (with the angle concluded by the polarization plane and e- equal to arg p ) ; right-handed elliptical: Im p>O; left-handed circular: p = -i; etc.

In the suggested projection, the real axis of the p plane should be chosen as the projec- tion of the circle L&Lj and the imaginary axis as the projection of the circle C L C , in Knitters Fig. 1.’ The origin coincides with L, and the scales are given by C2 = i, Ls = 1.

This author has found from relation (2) the most convenient way of expressing the polarization of a plane wave, mainly because it eliminates the troublesome confusion be- tween space vectors and time phasors (retain- ing only the latter3).

PETR BECKMANN Dept. of Elec. Engrg.

University of Colorado Boulder, Colo.

netic waves scattered from rough surfaces,” S p q . J P. beck ma^, “The depolarization of electromag-

Electromagnetic Theory and Anterutas (Copenhagen: June 1962). New York: Pergamon P m s , 1963. pp. 717-726.

Radiation Muence Coefficient Abstract-An expression is given for the

Schelkunoff-Friis radiation influence coefficient for arbitrary orientations of the current ele- ments with respect to the coordinate system.

By the methods of moments, the power P radiated by a discrete assemblage of current elements is given by

1 P = li,, cosO,,dU,,,du, (1)

- m n

Manuscript received April 7,1967.