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NtSSlR 81-1644

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AN APPROXIMATE EXPRESSION FOR THE PRINCIPAL

BEAMWIDTHS OF DIRECTIVE ANTENNAS

IN TERMS OF APERTURE FIELDS

Arthur D. Yaghjian

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

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81-1644

1981

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March 1981

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AN APPROXIMATE EXPRESSION FOR THE PRINCIPAL

BEAMWIDTHS OF DIRECTIVE ANTENNAS m

IN TERMS OF APERTURE FIELDS c 5

Arthur D. Yaghjian

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

March 1981

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U S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

An Approximate Expression for the Principal

Beamwidths of Directive Antennas in Terms of Aperture Fields

Arthur D. Yaghjian

An approximate, but general, formula for the half-power

prinicpal beamwidths of directive antennas is derived from

the Fourier transform of the aperture electric field. The

derivation, which can be accomplished by expanding the

transform in a Taylor series, is greatly streamlined by

introducing a Gaussian beam approximation. The formula,

which also relates beamwidth to directivity indirectly

through simple integrations of the aperture field, reduces

to a very compact expression in terms of the zero and second

order moments of the aperture distribution for linearly

polarized, uniformly phased aperture fields.

Key words: Antenna beamwidths; aperture fields; directive

antennas.

Because the far field of an arbitrary antenna can be expressed

exactly in terms of a double Fourier transform of the transverse

electric field on an "aperture" plane in front of the antenna [1], the

directive gain function of an antenna can also be expressed simply in

terms of the aperture transverse electric field E by the following

3

formula:

2

D (e.4>) = J2p

0

2

A A ^A ^ cos e (1-sec 6 e e •)/ E exp(-ike • R )dA . (la)

z r . a r a A

We will be concentrating on the main beam of directive antennas which

are electrically large (in both aperture dimensions) with fairly uniform
phase + for each significant polarization in the transverse aperture

field, thus producing a narrow main beam. (The term "directive" will be

used throughout to denote specifically this class of antennas.) In the

2 main beam of such antennas e

2 is much less than 1 radian , and (la) is

well approximated by the simplified formula, 1^

D (6 ,<J> ) -

2tt Y

( aV / E

a exp(-iker

. *a

)dA (lb)

The coordinates in (1) are defined in figure 1, k = 2tt/a is the freespace propagation constant, Ais the wavelength, Y Q

is the wave

admittance, A is the designated planar aperture which is truncated to a finite size, and exp(-iajt) time dependence for the fields has been

suppressed. If the power PQ

radiated by the antenna is approximated by

integrating ,5Y IE I 2 over the aperture plane, (1) reduces, for its

maximum value, to the familiar formula for directivity D

Q

in terms of

aperture fields ([la], eq (3.37)),

0 =

4Tr o X2

” i

E

. dA

El 2 a dA

( 2 )

The direction of maximum gain has been chosen to lie along the z-axis

(6 - 0), i.e., D

0 (O) 5 = 0.

The main purpose of this communication is to derive a simple

approximate formula, similar to (2) for half-power (3 dB) beamwidths of

directive antennas in terms of aperture fields. We can accomplish this

by expanding (1) in a Taylor series about e = 0, thereby generating a generalized moment expansion, and discarding terms which are negligible

for directive antennas. However, it turns out that the same basic

^This restriction can be relaxed for many aperture illuminations (e.g.,

see (16)), and the central result (10) remains a good approximation.

^Equation (lb) is a decent approximation to (la) for fairly large 0;

e.g., when e = 15 degrees, D(e,<f>) determined from (la) and (lb) differ

by only a few percent.

2
approximate formula (10) which results for principal beamwidths can be

derived in a much less complicated way by assuming the directive gain

within -3 dB of the main-beam axis obeys a Gaussian curve with respect

to 0; specifically,

(3)

(Far-field patterns determined from measurements indicate that (3) is

satisfied down to -3 dB to a high degree of accuracy for directive

antennas.

)

Observe in (3) that the beam is not assumed to be circularly

symmetric but can vary azimuthal ly with <j>. However, the azimuthal

function g((f>) is restricted by the fact that for finite-sized apertures

(1) demands all derivatives of D with respect to x and y be

continuous. Since x = r sine cos<j> and y

= r sine sin<j>, a straightforward application of the chain rule to any function D(e,<j>) with

continuous second derivatives proves that

where subscripts indicate their respective partial derivatives, and the

superscript o denotes the values of the partial derivatives D xx , D xy , and D at (x,y,z) = (0,0,1). D° , D° , and D° are constants for

yy xx xy yy

chosen xy axes; however, -D (0,<}>), the curvature of D ( e

,

4

> ) at e = 0,

u u varies, in general, as a function of <|>. For the sake of some simplification, we can choose the xy axes along principal directions defined so that D° = 0. (A straightforward

xy

rotation of coordinates proves such a choice is always possible [2].

For many linearly polarized antennas, these principal directions will

coincide with "E-plane" and "H-plane" cuts.) Noting from (3) that

we can then express g

(

4

) ) more specifically in terms of curvature as

D

0e

(O,<j)) = D®

x cos2

<j) + D°

y

si n2<j> + D^

v sin2

<j), ( 4 )

D 0e

( o »4>) - -2 D q

g(<(>). ( 5 )

3
gU) [D° ( 6 ) x cos 2

<|) + d°

yy

sin2

<j>]

A formal expression for the half-power beamwidth e^, which is a function of follows directly from ( 3 );

6 b

2 /inr • < 7 >

with, of course, g( 4

> ) given by ( 6 ). By inspection of (4) or ( 6 ) we obtain the elementary theorem of differential geometry [ 2 ], that the

principal curvatures along the principal directions ( 4) = 0 or it

)

and

(<j> = j

or y~) are the minimum and maximum curvatures (or vice versa).

Hence from (7) and ( 6 ) the "principal beamwidths" along the principal

directions are the minimum and maximum beamwidths given by

( 8a)

0 g = 2 / 2zn 2 ( 8 b)

Moveover,

principal

e g from (7) can be rewritten for all $ in terms of the

beamwidths from ( 8 );

e B (*>

x y

0 0 B B

( 0 g) si n

2

<j> + (

g

)

COS'

(9)

Equation (9) is the representation of an ellipse in polar coordinates,

and thus expresses explicitly that the -3 dB contour for narrow main

beams approximates an ellipse.

All that remains to finding the beamwidth in terms of aperture

fields is to evaluate the partial derivatives D° and D° from ( 1 ). xx yy

This straightforward exercise in calculus finally yields for directive

4
antennas,

B

( 10 )

For most directive antennas used in practice, exhibits a reasonable degree of even symmetry with respect to x and y axes whose

origin is located at a central point in the aperture. For these

antennas the second term in the denominator of (10) is negligible

because x

g and ya are odd functions with respect to the origin. For a perfectly symmetric (even with respect to x and y) aperture field, or for a linearly polarized, uniformly phased aperture field, the origin

can always be chosen to make this second term exactly zero, i.e.

dA = 0. (11a)

Recall also that the orientations of the xy axes have been chosen along

principal directions defined such that D° = 0, which for the above

xy

stated aperture fields satisfying (11a), translates from (lb) to

/

E x v dA = 0 . (lib)

A

a a a

If the transverse electric field is given or measured, an estimate

of the beamwidths in (8) and (9) is provided through evaluation of the

four (and usually three) independent aperture area integrations in

(10). For linearly polarized, uniformly phased aperture fields, (10)

and (11) reduce to a more compact form which allows e* and from (8)

to be written in convenient "moment" terminology,

e x

B

2 /"IrTT

k

5

(12a)
(12b) y _ 2 / in 2

' B J-—

(m 10 m 01

" y ll

0)

where u mn

is the joint moment defined by

^mn 1 m n . . x y dx dy . (13)

X V c For a circularly symmetric beam, 6 g = e g = e g and (12) combine with (9)

to produce

Q c _ 2 /2an 2

e B R

' 1A

E

a dA

1

A

E

a R2 dAa

(14)

Equations (10), (12), and (14) are approximate expressions for the

beamwidth which approach in simplicity of form and ease of evaluation

the corresponding approximate expression (2) for directivity. Although

(10), (12), and (14) were derived by assuming a Gaussian main beam

pattern down to -3 dB, it is emphasized that this is an unnecessary

artifact introduced to greatly streamline their derivation. The same equations except for a factor Jl ~ \/T~ = 0.77 replacing / £ n 2 = 0.83

emerge directly from (1) by expanding /D ( 0 ,<pf in a Taylor series about

6=0, retaining terms out to the fourth power in 6, setting B

0 (O) = 0,

and showing that the third and fourth terms are negligi ble for directive

antennas. If the /in 2 factor in (12) is replaced by Jl - /T~, (12)

becomes identical to Spencer's result for two-dimensional scalar

patterns from uniform-phase, symmetric aperture distributions ([3],

Sec. 6.6).

Table I compares beamwidths computed from the approximate

expression (14) with the exact values obtained by Hansen [4] and Silver

([3], Table 6.2) for circular aperture distributions with different

amplitude tapers. The maximum discrepancy of 3 percent between exact

and approximate values occurs for the uniform distribution (no taper).

6
Finally, we estimate 0g for the same circular aperture

distributions using the common, rule-of-thumb relationship [5] which

emerges between directivity and principal beamwidths when the total

radiated power is roughly approximated by the power in the "equivalent"

Gaussian beam (3). Specifically, insert D(e,<|>) from (3) into the

identity,

, 2lJ 7T 1 =-4^- / /

D(e,<f>) sine de d<j>, 0 0

perform the integration for large g(<f>) (narrow beams), and obtain after

substitution from (8)

°o

16 in 2 11.09

e r %

eB0 B

(15)

This relationship, which has been derived by Ko [6] for circular beams

( 0 B

= e Er*

re P resents somewhat of a compromise between Kraus's

formula [5], D = 4tt

/

ej 0^ , and Tai and Pereira's formula [7],

0 D D D

0

* 16 in 2 / 1/2 (6g

2 + )

•

The values of 0 g calculated from (15) (with DQevaluated from (2)) are included in table I. On the average they do not lie as close to the

exact beamwidths as the values calculated from (14). But this would be

expected since the derivation of (15), unlike (14), depends strongly

upon the crude approximation that the total radiated power is contained

in an equivalent Gaussian beam. For antennas with pattern shape factors

or beam efficiencies differing appreciably from unity [5b], the

relationship (15) between directivity and principal beamwidths must be

modified by these parameters to remain useful, whereas (10) continues to

yield a reliable estimate of beamwidths because it inherently accounts

for variations in beam efficiencies and pattern shape factors. This is

a distinct advantage of (10) because often pattern shape factors and

beam efficiencies are not readily available.

As a simple yet graphic example of an antenna aperture distribution

7
for which (15) fails but (10) gives an accurate approximation to the

beamwidths, consider the electrically-! arge square aperture of side S

illuminated with a linearly y-polarized transverse electric field

Ey

= 1 + 3e

lkxsina

, (Q<3<1, <a< ~). (16)

Such an antenna would have its mainbeam near 0 = 0 and a grating lobe near e = a. Its mainbeam is nearly circular with beamwidth and

directivity given approximately by .89A/S and 4ttS2 /A 2 (1 + 3 2 cosa),

respectively. Thus, the beamwidth and directivity of this antenna

greatly deviate from the relationship (15) for non-negl igible 3 2 cosa. However, application of the formula (10) yields approximate beamwidths,

6g * 0^ - .92 A/S,

which are within a few percent of the actual beamwidths.

8
TABLE 1

Exact and approximate values of beamwidth for circular aperture

distributions {b + [1 - (R /a) 2

] n }. (a is the radius of the aperture.)

a

Beamwidth x ~

n exact approximate

eq. (14)

approximate

eq. (15)

o

II £.1

0 1.03 1.06 1.06

1 1.27 1.30 1.22

2 1.47 1.50 1.43

3 1.65 1.68 1.58

4 1.81 1.84 1.77

b =1/4

1 1.17 1.20 1.14

2 1.23 1.25 1.18

3 1.25 1.27 1.19

b =V2

1 1.13 1.162 1.105

2 1.16 1.186 1.130

3 1.16 1.186 1.136

9
y

Figure 1. Definition of coordinates for

far-field and aperture integration;

R

a x e + y e . ax J a y

10
REFERENCES

[1] a) Collin, R.E., and Zucker, F.J., Antenna Theory, part I, ch. 3.

McGraw-Hill, New York (1969).

b) Kerns, D.M., Plane-wave scattering-matrix theory of antennas

and antenna-antenna interactions: formulation and

applications, J. of Res. Nat. Bur. of Stand., vol . 80B, pp.

5-51 (Jan. -Mar. 1976).

[2] Gauss, K.F., General Investigation of Curved Surfaces, ch. 1, sec.

8, Raven Press, New York (1965) (translated from the Latin and

German by A. Hiltebeitel and J. Morehead).

[3] Silver, S., Microwave Antenna Theory and Design, ch. 6, Dover

Publications, New York (1965) (first published by McGraw-Hill in

1949).

[4] Hansen, R.C., Microwave Scanning Antennas, _I_, ch. 1, Academic

Press, New York (1966).

[5] a) Kraus, J.D., Antennas, eq. 2-45, McGraw-Hill, New York (1950).

b) Radio Astronomy, eq. 6-22, McGraw-Hill, New York (1966).

[6] Ko, H.C., Radio telescope antenna parameters, IEEE Trans. Mil.

Electron., MIL-8, pp. 225-232 ( July-October 1964).

[7] Tai, C.T., and Pereira, C.S., An approximate formula for

calculating the directivity of an antenna, IEEE Trans. Ant.

Propagat., AP-24, pp. 235-236 (March 1976).

11
NBS-1 14A (REV. 2-8C) U.S. DEPT. OF COMM. 1. PUBLICATION OR j 2. Performing Organ. Report No. 3. Publication Date BIBLIOGRAPHIC DATA

SHEET (See instructions)

REPORT NO.

NBSIR 81-1644 March 1981

4. TITLE AND SUBTITLE

An Approximate Expression for the Principal Beamwidths of Directive Antennas in

Terms of Aperture Fields

5. AUTHOR(S)

Arthur D. Yaghjian

6. PERFORMING ORGANIZATION (If Joint or other than NBS, see in stru ction s) 7 . Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

10. SUPPLEMENTARY NOTES

[22] Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here) An approximate, but general, formula for the half-power principal beamwidths of

directive antennas is derived from the Fourier transform of the aperture electric

field. The derivation, which can be accomplished by expanding the transform in a Taylor series, is greatly streamlined by introducing a Gaussian beam approximation.

The formula which also relates beamwidth to directivity indirectly through simple

integrations of the aperture field, reduces to a very compact expression in terms of

the zero and second order moments of the aperture distribution for linearly polarized

uniformly phased aperture fields.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) Antenna beamwidths; aperture

13. AVAILABILITY

fields; directive antennas.

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