Optimum Yagi Joined

Orfanidis tiene una funcion llamada yagi, que calcula:

1. Las corrientes de entrada de los dipolos, se consideran dipolos el reflector, los directores y eldipolo propiamente dicho,

2. La directividad en veces

3. La relacion adelante atras en veces

y cuyos parametros de entrada son tres arreglos unidimensionales:

1. La Longitud de los dipolos en lambdas

2. El radio de los alambres en que se construyen los dipolos, en lambdas

3. La ubcacion de los dipolos a lo largo del eje x, seguramente en lambas pero no dice nada ladocumentacion, probar esto y consultar codigo fuente para estar seguros

Si con los resultado obtenidos se utiliza la funcion gain2d se puede obtener un grafico de la direc-tividad, ojo que la documentacion habla de la funcion array2d pero no es as.

Debe presentar un documento .m donde:

1. Sustente la parte del codigo fuente de yagi.m donde se aclare si la distancia entre los dipolosesta dada en metros, lambdas o que unidad.

2. Realice el ejemplo ultimo dgito de su cedula %3 + 1 del artculo: Cheng, D.; Chen, C., Op-timum element spacings for Yagi-Uda arrays. Antennas and Propagation, IEEE Transactions on, vol.21, no.5, pp.615,623, Sep 1973. Los ejemplos se encuentran en la seccion VII NumericalExamples.

Nota: % es el operador que calcula el residuo de una division entera. Por ejemplo si el ultimodgito de su cedula es 6 entonces 6 %3 = 0 + 1 = 1, debe hacer el ejemplo 1.

3. Compare los resultados obtenidos por el autor en 1973 con las herramientas que tenemos yagi.my gain2d.m. Si piensa que no se pueden obtener resultados con estas herramientas justifique.

4. grafique los patrones de radiacion, para el ejemplo que le correspondio de los arreglos inicial yoptimizado.

5. Bono obtenga los graficos que se presentan en el artculo: Fig. 2. y Fig. 4. El ejemplo 3 noesta graficado.

1

IEEE TRANSACIIONS ON ANTENNAS AND PROPAGATION, VOL. AP-21, NO. 5, SEPTEMBER 1973 615

Optimum Element Spacings for Yagi-Uda Arrays DAVID K. CHENG AND C. A. CHEN

Abstract-A method is developed for the maximization of the forward gain of a Yagi-Uda array by adjusting the interelement spacings. The effects of a bite dipole radius and the mutual coupling between the elements are taken into consideration. Currents in the array elements are approximated by three-term expansions with complex coefficients which convert the governing integral equations into simultaneous algebraic equations. The array gain is maxi- mized by the repeated application of a perturbation procedure which converges rapidly to yield a set of optimum, generally unequal, element spacings. This method eliminates the need for a haphazard trial-and-error approach or for interpreting a vast data collection. Illustrative examples are given.

B I. INTRODUCTION

ECAUSE of t,heir simplicity and versatility, Yagi- Uda antenna a.rrays [l], [a] have found many

important practical a.pplicat,ions. A conventional Yagi- Uda array consists of a row of para.lle1 straight cylindrical dipoles, of which only the second one is driven by a source and a.11 others are parasitic. Fig. 1 represents a typical arrangement. The driven element no. 2 is norma.lly tuned to resonance. Element no. 1 is a reflector which is usually longer than the driven element, while elements no. 3 to N are directors and are shorter than the driven element.

An important performance index of Yagi-Uda arrays is the gain, or directivity if element losses are neglected. Arra.y directivity depends on the radius and length of the dipole element,s as well as on the total number of elements and the spacings between them. Theoretically, each of those pa.rameters can be varied individually, making the problem of finding an absolute optimum combination practically impossible. Attempts have been made by various investigators t,o maximize the gain of Yagi-Uda arrays [3)-[5]. However, the result,s have not been meaningful on account of inherent rough approximations.

Ehrenspeck and Poehler [SI examined experimentally and systemat,ically, a method for obtaining maximum ga,in from a Yagi-Uda array with equally spaced directors of equal length. They concluded that the phase velocity of the surface wa,ve traveling along the row of directors could be used as a. design criterion. This phase velocity depends, of course, on the element length and spacing paramet.ers in a very complicated way. The surfa.ce-wave concept has also been used to calculate the phase velocity of infinitely long uniform dipole arrays with assumed current distribut,ions [7], [SI, to determine cutoff fre- quencies [9] and choose design parameters [lo 1, and t o analyze long Yagi-Uda, arrays [ll].

1973.

Engineering, Syracuse Univermty, Syracuse, N.Y. 13210.

Manuscript received December 27, 1972; revised February 23,

The aut6ore are with the Department of Electrical and Computer

U 1 2 3 1 L N

Fig. 1. Typical Yagi-Uda array.

Recently, Bojsen et al. [12] used a numerical approach to obtain curves showing -the variation of maximum gain with element spacing for endike half-wave dipole arrays, of which only one element is excited and the rest are parasitics center-loaded with reactances. Sinusoidal current distributions were assumed and the effects of dipole radius were neglected. Their results appear to dispute the propriety of basing the optimum design solely on traveling- wave considera,tions. The nonuniformity in both the ampli- tudes and the progressive phase shifts of the currents in the elemenk of a Yagi-Uda array has been pointed out by Thiele [13].

Morris [14] used a three-term approximation for antenna currents to solve the governing simultaneous integral equations for Yagi-Uda arrays. He obtained a large amount of interesting da$a for typical arrays nith 2, 4, and 8 equally spaced directors of equal length. Of partic- ular sigficance is the evidence that the array gain drops sharply when the director spacing is larger than about 0.4X, which confirms the findings of Ehrenspeck and Poehler [SI.

The purposes of this paper are to demonstrate that the forward gain can be increased by spacing the directors . nonuniformly and to present a method for determining the spacings needed for gain optimization. The method employs a spacing perturbation technique [15] which converges quite rapidly. The three-term theory developed by King and his associates [lS] is used t o convert the integra.1 equations into a set of algebraic equations. Typical numerical results are presented, and radiation patterns and current distributions on the array elements are plotted.

The spacing perturbation technique could be used in conjunction with the method of moments [17] which also converts integral equations into simultaneous algebraic equations. However, in subsectioning the array elements, matrices of much larger dimensions would have to be manipulated. This limits the number of array elements that can be handled, Furthermore, the currents on the pa.rasitic elements depend much more critically upon those on other mutually coupled elements. The effects of small errors multiply and there would be convergence problems

616 IEEE TRANSACTIONS ON ANTEXNAS AND PROPAGATION, SEPTEWBER 1973

unless more subsections than those normally required for driven elements are taken. In using the three-term theory the largest matrices to be handled for an N-element array are of a dimension N X N and no convergence problems are encountered.

11. CURRENT DISTRIBUTIONS IN YAGI-UDA ARRAY

We shall s u m m h e in this section the integral-equation formulation for the currents in the N elements of an Yagi-Uda array using a three-term approximation for the driven element and two terms with complex coefficients for the para.sitic elements [E] . The N simultaneous integral equations to be solved are

N hi 2 /-hi li (Xi ') Kkia ( z k , z i ' ) dzf

(3)

Rki ( z k ) = [ ( z k - z / ) 2 + bk?]'" (4) R k i ( h k ) = [ ( h k - Z ( ) z + bki2]1'2 (5)

b k k = a. ( 6 )

In solving the simultaneous integral equations (1) , the current distributions l i (2) are assumed to have the follow- ing form:

a l i ( 2 ) = Ai(m))Xi(m) ( 2 ) (7)

m=l

with Xi("(2) = sin/9O(hi - I z I) (8) Xi'') ( z ) = COS / 3 ~ - COS &hi (9) Xi(3) ( x ) = cos +/3& - cos +pohi (10)

m d AJl) = 0 for i # 2 (parasitic elements). Substitution of (7) in (1) and use of certain approximate relations for the integrals involved yield

and two simultaneous matrix equations for the column matrices of complex coefficients ( A ( 2 ) ) and {A(3)} :

[ W ) ] { A ( 2 ) ) + [@~(3)]{A(~)j = - (@~( l )JA2( l ) (12) [XPd(2)]{A(2)} + [9d (3 ) ] [A(3 ) } = - { \E&)}Az(~) . (13)

The expression for *=a(') and the elements of the AT X 1 column matrices { @z(l) ) and {9z,J1) ) as well as those for the N X N square matrices [ W ] , [ W ] , [9d(2)], and [ @ P ] a,re rather involved. They can be found in [16) and [ls]. It is noted that when the geometrical dimensions are known, the complex coefficients Ad'), {A(?) 1, and { A @ ) ) can be eva,luated from (11)-(13). From (7), the current distributions in all the elements of a Yagi-Uda array can then be determined.

When the driven element (no. 2) in a Yagi-Uda array is a half-wave dipole, as it often is, &hz = r/2 and cos @oh2 = 0. Some of the quantities in the preceding formulation mill become indeterminate. Although they yield definite values in the l i t i n g process, an alternative formulation is preferred in order to avoid computational dif6culties. The integral equation (1) for the driven half- wave dipole becomes [18]

N hi 5 1 i ( 2 i ' ) ~ 2 S ( e , Z l ) dz; - - 30 [iVoz(~in Bo I z 1 - 1) + Oz COS 602) (14) - j

with N h i

0 2 = j30 / I ; ( z + 1 - cos - 1 - sm - . (20) [ C)Il

CHENG AND CHEN: SPACINGS FOR YAGI-UDA AFtRAYS 617

k P 2

k = 2

--here

The elenlents of [ ! @ d @ ) ] and [ + d c 3 ) ] in (19) are the same as those for [ \ k J m ) ] in (13) for m = 2 and 3. The elements of [6(2)] and [ & ( 3 ) ] are the same as those for [ + c 2 ) ] and [W] except when k = 2. For k = 2, we have

&(2) = I p Z i ( 2 ) (0) - ip2id'(2) (26)

&(3) = qrti(3) (0) - q r 2 & p 3 ) (27)

where f P 2 i ( 2 ) ( 0 ) and \k2i(3) (0) are

ezi(")(0) = 1 Si(m)(zi')Kzi(O,z() dzi', m = 2,3. (28) The current distributions in the e1ement.s of a Yagi-

Uda array m

618 IEEE TF~ANSACIXONS ON ANTENNAS AND PROPAGATION, s m m ~ 1973

The expressions for the newly defined complex coefficients we have 4,Jm), Ijkkid(m), 4kZ( l ) , and yiM(1) are given in Appendix I.

in (7) will also be changed. We write I i d ( 3 ? 1 ) = [ ['@')] [ @ d @ ) l l I C F J { Ad} The coefficients {A@) ) and { A(3) } for the current terms { Ai4- ('1 } [ @ ) ] [ 6 ( 3 ) ] -1 [Fz] { A ( Z ) j P = { A @ ) } + { b A ( Z ) } (39) {AW'jp = {A(3)) + { A i 4 ( 3 ) ) . (40) (46)

Substitution of the perturbed matrices (31)-(34), (39), and (40) into (12) and (13) yields, when second-order IV. R A D I A ~ O N FIELD FROM PEETURBED ARRdy deviation terms are neglected: The radiation field of a spacing-perturbed linear array [O(Z)]{AA(2)} + [9(3)-J{ A A ( 3 ) } at a distance Ro from a reference origin is

(42) For small Ad;, that is; for

In view of (35)-(38), the kth element of the right-hand ( Adi) /di

CaENG AND CHEN: SPACINGS FOR YAGI-UDA ARRAYS 619

With (51), we can mite (48) as

E'(4dJ) = E(&+) + a ( e , $ ) = E(4+) + DIT{Adl (52)

where the superscript T denotes transposition. Equation (52) is useful because the deviation field AE due to spacing perturbation is expressed explicitly in terms of {Ad}. We are now ready Do consider the problem of gain optimization of Yagi-Uda arrays by spacing perturbation.

V. GAIN OPTIMIZATION BY SPACING PERTURBATION The gain of an a.rray in the direction (eo,&) is

(53)

where Pi, is the t,ime-average input power. With spacing perturbation E becomes E', Pi, becomes Pin', and the perturbed gain becomes

(54)

From (52),

I Evo,do) 12 = (E* + AE*) (E' + AE) = I E 1' + 2{Ad)T{B~] + {Ad]'[Cl]{Ad]

(55)

{BI) = Re (E'{D*}) (56) where

and CCll = { D } * { D } T . (57)

In (56), Re ( E { D*) ) = real part of the product of E a,nd the complex conjugate of the column matrix ID}. The AT X N square matrix [C,] is positive semi-definite, and, since {Ad] is a real mat,rix, [Cl] in the last term of ( 5 5 ) can be replaced by m e Cl]. Pi,,' in (54) is

Pin' = $ R e [Voz*IzP(O)] = Pi, + (Ad)T{B2) (58) where p . In - - - :Va Re [A2(')&(')(0)

+ A z ( ~ ) S Z ( ~ ) (0) + A z ( ~ ) X ~ ( ~ ) (0) 1 (59) and the kth element of the column matrix (B2} is

For a. lossless array, the input power equals the total power ra,diated, a.nd Pin' can be mitten in an alternative form :

Using (52), (61) ca.n be expressed ax

Pi,' = Pi, + 2{Ad}T{B3} 4- {Ad}T[Re Cz]{Ad} (62) where the radiated power of the unperturbed array

i r2a

and i r2* r z

{ B I } and [Cl] have previously been defined, respectively, in (56) and (57) ; and [CZ] is a positive definite Hermitian matrix.

The objective of gain optimization by spacing perturbation is t o find the small changes in the element spacings such thak the array gain in a. given direction is increased a.nd to repeat the process until further increases in gain are negligible. Hence, it is essential that

AG(eo,do) = G'(eo,b) - G(@o,+o) (66)

be positive. Substitution of (53)- (62) in (66) yields

Note that the negative sign in (68) for { B } in the numer- ator of AG(Bo,dJo) in (67) implies that the array gain d decrease for an improper choice of { Ad}.

In order t o be certain that AG(&,&) slcill be positive, we make use of a known relation in the theory of matrices [15], [19]. Applied to the present problem, the relation asserts that if m e Cz] is positive definite, then

({BITIRe C2T1(B])({AdIT[Re CZ]{A~})

2 ({Ad)T(B])2. (69)

In (69), the equality sign holds when

{ A d ) = a[Re CZ]-~{B} (70)

where CY is a positive consta.nt. Hence, if t>he spacing changes in { A d } are chosen, such that,

{Ad] = a[Re C2]-1(2{B~] - 60G{B2)) (71)

then

620 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, SEPTENBER 1973

TABLE I GAIN OPTIMIZATION FOR SIX-ELEMENT YAGI-UDA ~ A Y

(PERTURBATION OF DECECMR SPACINGS) 2h1 = 0.51X, 2h9 = 0.50% 283 = 2h4 = 2h5 = 2hs = 0.43X,

a = 0.003369X

InitiSlArray 0.250 0.310 0.310 0.310 0.310 8.06 Optimizedhray 0.250 0.336 0.398 0.310 0.407 11.81

0 . 5 - c - 0.4-

0.3

0.2

0.1

-

-

-

0 15 30 45 60 75 90 105 120 135 150 165 160 # DEGREES

Fig. 2. Normalized patterns of six-element Yagi-Uda arrap (Example 1).

TABLE II GAIN OPTWATTION FOR SIX-ELEMENT YAGI-UDA ARRAY

(PERTURBATTON OF ALL ELEXENT SPACINGS)

b2dX b d X b43/X b d X b d X Gain

Initial Array Optimized Array 0.250 0.352 0.355 0.354 0.373 11.85

0.280 0.310 0.310 0.310 0.310 7.53

The a in (71) should be sufficiently small to sat,isfy the condition ( Adi) / d i

CHENG AND CHEN: SPACINGS FOR YAGI-UDA ARRAYS 621

TABLE III GAIN OPTIMIZATION FOR TEN-WNT YAGI-UDA AILRAY

(PERTORBATION OF D~~ECTOR SPACINGS) 2hl = 0.51X, 2hZ = 0.50X, 2hi = 0.43X (i = 3, 4, .**, lo), ~a = 0.003369X

bl/x b d X b,,/X b d X b6dX b16/h bm/X b p d X blO.O/X Gain

Initial Array Optimized Array

0.250 0.330 0,330 0.330 0.330 0.330 0.330 0.330 0.330 12.36 0.250 0.319 0.357 0.326 0.400 0.343 0.320 0.355 0.397 16.20

for t.he opt,imum array has not only a narrower main beam but also lower sidelobes, a fact. which has been noted previously [20].

Example 2: Six-element, Yagi-Uda array 1it.h a half- wave driver (2h2 = 0.50X; one reflector, 2hl = 0.5lX; four directors, 2h3 = 2hl = 2h5 = 2$ = 0.43X; a = 0.003369h). In the initial a,rray, b ~ 1 = 0.280X, b32 = b.13 = b s = bG5 = 0.310X. All element spacings a.re to be adjusted for gain optimization.

The reflector spacing bP1 in the initial srray is arbitrar- ily chosen to be 0.280X, and all ot,her element spacings are given as 0.310X. The gain of this initial array is 7.53 (8.77 dB). Now all element spacings are adjusted simul- taneously in t,he optimization procedure t,o increase t.he gain. The results are summarized in Table 11. The gain of the optimized array is 11.85 (10.74 dB), an increase of 57.3 percent (1.97 dB).

The real and imaginary pa,rts of t.he currents in the elements of t.he optimized array with unequal spacings are plot,ted in Fig. 3, which includes t,he effects of mutual coupling and finite dipole radius. It is interest,ing to find that, the director spacing for t,he optimized array is 0.2501, which confirms with what has been found by other investigators [lS]. The normalized radiation patterns for both the initial and t,he optimized arrays are given in Fig. 4. Again, the pat.t>ern for t,he optimized a.rray has a narrower main beam as well as lower sidelobes. The computed relative field intensities in the direction of ma.ximum radia,t,ion are 0.920 and 0.976, respectively, for t,he initial a.nd optimized arrays.

Example 3: Ten-element Yag-Uda a.rray xith a half- wa.ve driver (2ha = 0.5OX; one reflector, 2hl = 0.5lX; eight directors, 2hi = 0.43X, i = 3,4, . . - , lo; a = 0.003369h). In the init,ial array, bPl = 0.250X, bB = b43 = = b10,9 = 0.3101. The director spacings are t.0 be adjusted for gain maximization.

Wit,h t.en elements in a Yagi-Uda array, it mould be impract,ical to use the moment method wit.h subsectioning for numerical solut-ion. However, only 10 X 10 matrices a.re involved in t,he present formulation. The results for the opt,imized array are summarized in Table 111. The calculated gain for t.he array with eight equa.lly spaced directors is 12.36 (10.92 dB) which checks very closely n-ith the result of hIorris [14]. The gain of the optimized array is 16.20 (12.10 dB, an increase of 31 percent (1.18 dB). Even for this example, the t,otal comput,ing time for seven iterat,ions on an IBM 370/155 computer t,ook only about 5 min.

VIII. CONCLUSIOK A method has been developed for the maximization of

t.he forward gain of a I'agi-Uda array by adjusting the interelement spa.cings. The effects of a finite element radius and the mutual coupling beheen t.he array elements are taken into consideration. A three-term expansion with complex coefficients is used to approximate the current distribution in the elements and to convert the governing integral equations int.0 simultaneous algebraic equations. The array gain is ma.ximized by the repeated application of a perturbation procedure which converges rapidly to yield a set, of opt,imum, generally unequal, element spacings. 1llust.rative examples are given to show typical gain increases that a.re attaina.ble with this technique.

Although the formulat,ion using t,he three-term theory a.ppears tedious, t.he end result is in a fairly simple form. The matrix equations need not be reformulated for differ- ent a.rrays once they have been obt.ained. The formulation itself is on firm grounds and has been expounded in many research a.rticles and several books. As far a.s its application to t,he present problem is concerned, the only numer- ically tedious part is the evaluation of definite integrals of the type given in (24), (25), (A-7), and (A-8). The method of moments with subsectioning cannot conven- ient.ly be used here because the critica.1 dependence of the currents in the parasitic elements on mutual coupling demands h e subsectmiorling and the consequent manipu- lat,ion of complex matrices of very large dimensions.

The largest matrices encountered in the spacing perturbation technique using the t,hree-term theory are of a dimension X X N for an LY-element array. The convergent iterat.ive procedure yields the opt.imum spacings for maximum gain nit.hout the need for a haphazard trial-and- error approach or for interpreting a vast dat,a collection.

APPENDIX I Expressions for 4drn) , 4kk2(1), $ d m ) , and $kBd( l ) in (35)-

(38) :

[ ; ; y l L k ) - $kid'(%) COSPOh.k, k # i =

k = i

(-4-1) $kZ(') ( h . k ) - (1 - 6k2) $k~a ' ( ' ) COS p&, k # 2

k = 2 +&) =

(A-2)

622 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, SEPTEMBER 1973

tities represent k # 2 (A- 14)

A2 = [l - COS B&k] [COS r$) - COS r$)] - [COS r$) - cos Boh.k ] [ 1 - cos r*)] (A-6) (A-7)

and

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATIOR, VOL. -21, NO. 5, SEPTEMBER 1973 623

c11

c21

c3 1 c41

C5 1 C6 1

REFERENCES

vol. 16, pp. 715-741, June 1928. H. Yagi, Beam transmission of ultra short wava, Proc. IRE,

Maruzen Co., 1954, S. Uda and Y. Mushiake, Yagi-Udu Antenna. Tokyo, Japan:

W. Walkinshaw, Theoretical treatment of short Yagi aerials, J . Inst. Ekc. Eng., vol. 93, pt. IIIA, no. 3, pp. 598-614, 1946.

Eke. Eng., vol. 93, pt. IIIA, no. 3, pp. 564-566, 1946. D. G. Reid, The gain of an idealized Yagi array, J . Inst. R. M. Fishenden and E. R. Wiblin, Design of Yagi aerials, Proe. Inst. Elee. Eng., vol. 96, pp. 5-12, Mar. 1949. H. W. Ehrenspeck and H. Poehler, A new method for obtain-

ing maximum gain for Yagi antennas, IRE Trans. Antennas Prupagai., vol. AP-7, pp. 379-386, Oct. 1959.

[7] D. L. Sengupta, On %he phase velocity of wave propagation along an idb i te Yagj structure, f R E Trans. Antennas

[SI F. Serracchioli and C. A. Lev& The calculated phase velocity Propagai., vol. AP-7, pp. 234239, July 1959.

of long end-fire dipole arrays, IRE Trans. Antennas Propagat., vol. AP-7, pp. S42443434, Dec. 1959.

[9] L. C. She:: Characteristm of propagat.ing waves on Yagi-Uda structure. IEEE Trans. Micruwave Thww Tech.. vol. MTT-

_ . .

19, pp,., 536-542, June 1971.

band Yagi arrays, IEEE .Trans. Antennas Propagat. (Com- , Directivity and bandwidth of single-band and double-

mun.), vol. AP-20, pp. 778-780, Nov. 1972. R. J. Mailloux, The long Yagi-Uda array, IEEE Trans. Antmnm Propagat., voL AP-14, pp. 128-137, Mar. 1966.

Andersen, Ivlax1IIO(um gam of Yagi-Uda arrays, Electron. J. H. Bojsen, H. Schaer-Jacobsen, E. Nilsson, and J. B. Lett., vol. 7, no. 18, pp. 531-532, Se t. 9, 1971. G. A. Thiele, Analysis of Yagi-&a-type antennas, IEEE Trans. Antennas Propagat., vol. AP-17, pp. 24-31, Jan. 1969. I. L. Morris, Optimizationofthe Yagi.Ama.y, Ph.D. dissertation, Harvard Univ., Cambridge, Mass., 1965. F. I. Tseng and D. I(. Cheng, Spacing perturbation techniques for array optimization, R d w Sn., vol. 3 (New Series), pp.

-

451-457; &fay 1968. . R. W. P. King, R. E. Mack, and S. S. Sandler, Arrays of Cylindrical Dipoles. New York: Cambridge U&v. Prees, 1968. R. F. Harrineton. Field Comvutation bu Mument Meulods.

_ _

New York: Ivl~cmihn, 1968. a D. K. Cheng and C. A. Chen, Optimum element spacings for Yagi-Uda arrays, Syracuse Univ., Syracuse, N.Y., Tech.

D. K. Cheng, Optimization techniques for antenna arrays, Rep. TR-72-9, Nov. 1972.

Proc. IEEE, vol. 59, pp.. 1664-1674, Dec. 1971. D. L. Sengupta, On d o r m and linearly tapered long Yagi antennas, IRE Trans. Antennas Propagat., vol. AP-11, pp. 11-17, Jan. 1960.

A New Method for Calculating Correction Factors for Near-Field Gain Measurements

ARTHUR C. LUDWIG AND RICHARD A. NORMAN

Absfract-A new method is presented for calculating near-field antenna gain correction factors directly from measured far-field pattern data by using a spherical wave expansion of the pattern. This eliminates the need for any assumptions regarding antenna aperture field distributions. The only significant assumption in the new method is to neglect multiple scattering between the antennas. The method is applied to the case of a horn antenna. Calculated results are compared to direct measured results, demonstrating agreement to within 0.03 dB. The method is also compared to the method of Chu and Semplak, with similar agreement. The sensitivity of the results to truncation error and noise in the data is also investigated and contrasted to sensitivity of prior methods to errors in the assumed field distribution.

work was supported by NASA under Contract NAS 7-100.

Institute of Technology, Pasadena, Calif. 91103.

Manuscript received January 29, 1973; revised April 2, 1973. This

The authors are with the Jet Propulsion Laboratory, California

I I. INTRODUCTION

T IS well lrnonm t,hat, t,he apparent gain of t.wo an- t,ennas separated by a finitme distance differs from t,he

gain in the limiting case of infinite separat.ion, and many authors have dealt m-it,h the problem of correcting for this effect [1]-[7]. All of t,hese prior techniques are analytical and typically involve an assumption that the fields are known to have a specific analytic form on some surface. Even t,hough t,he results generally agree very well with experimental dat,a, it, is difficult to assign an exact tolerance to the computed correction factors due to the various assumptions used [8], [9].

It is the purpose of this paper to present an a.lt,ernate a,pproach based on the use of experimental data, rat,her than an assumed field distribut,ion. This method will be

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Optimum Yagi Joined