Because a ratio of radiation intensities is used to calculate directivity, the pattern may bereferred to any convenient level. The most accurate estimate is based on measurementsat equal angle increments over the entire radiation sphere. The average may be foundfrom coarse measurements by using numerical integration, but the directivity measuredis affected directly by whether the maximum is found. The directivity of antennas withwell-behaved patterns can be estimated from one or two patterns. Either the integralover the pattern is approximated or the pattern is approximated with a function whoseintegral is found exactly.

By estimating the integral, Kraus [4] devised a method for pencil beam patterns withits peak at θ = 0â—¦. Given the half-power beamwidths of the principal plane patterns, theintegral is approximately the product of the beamwidths. This idea comes from circuittheory, where the integral of a time pulse is approximately the pulse width (3-dBpoints) times the pulse peak: U0 = θ1θ2/4π, where θ1 and θ2 are the 3-dB beamwidths,in radians, of the principal plane patterns:

directivity = (rad) = (deg) (1)

Example Estimate the directivity of an antenna with E- and H-plane (principal plane)pattern beamwidths of 24â—¦ and 36â—¦.

Directivity = = 47.75 (16.8 dB)

An analytical function, cos^2N(θ/2), approximates a broad pattern centered on θ = 0â—¦with a null at θ = 180â—¦:

U(θ) = cos^2N(θ/2) or E = cos^N(θ/2)

The directivity of this pattern can be computed exactly. The characteristics of theapproximation are related to the beamwidth at a specified level, Lvl(dB):

beamwidth [Lvl(dB)] = 4 cos⁻¹(10^{−Lvl(dB)/20N}) (2a)

N = (2b)

directivity = N + 1 (ratio) (2c)

SCALE 1 3-dB beamwidth and directivity relationship for cos2N(θ/2) pattern.

SCALE 2 10-dB beamwidth and directivity relationship for cos2N(θ/2) pattern.

Scales 1 and 2, which give the relationship between beamwidth and directivityusing Eq. (2), are useful for quick conversion between the two properties. You canuse the two scales to estimate the 10-dB beamwidth given the 3-dB beamwidth. Forexample, an antenna with a 90â—¦ 3-dB beamwidth has a directivity of about 7.3 dB. Youread from the lower scale that an antenna with 7.3-dB directivity has a 159.5â—¦ 10-dBbeamwidth. Another simple way to determine the beamwidths at different pattern levelsis the square-root factor approximation:

 =

By this factor, beamwidth_{10 dB} = 1.826 beamwidth_{3 dB}; an antenna with a 90â—¦ 3-dBbeamwidth has a (1.826)90â—¦ = 164.3â—¦ 10-dB beamwidth.

This pattern approximation requires equal principal plane beamwidths, but we usean elliptical approximation with unequal beamwidths:

U(θ,φ) = cos^2Ne (θ/2) cos² φ + cos^2Nh(θ/2) sin²φ (3)

where Ne and Nh are found from the principal plane beamwidths. We combine thedirectivities calculated in the principal planes by the simple formula

directivity (ratio) = (4)

Example Estimate the directivity of an antenna with E- and H-plane pattern beamwidthsof 98â—¦ and 140â—¦.

From the scale we read a directivity of 6.6 dB in the E-plane and 4.37dB in theH-plane. We convert these to ratios and apply Eq. (4):

directivity (ratio) = = 3.426 or 10 log(3.426) = 5.35 dB

Many analyses of paraboloidal reflectors use a feed pattern approximation limitedto the front hemisphere with a zero pattern in the back hemisphere:

U(θ) = cos^2N θ or E = cos^N θ for θ ≤ π/2(90â—¦)

The directivity of this pattern can be found exactly, and the characteristics of theapproximation are

beamwidth [Lvl(dB)] = 2 cos⁻¹(10^{−Lvl(dB)/20N}) (5a)

N = (5b)

directivity = 2(2N + 1) (ratio) (5c)

We use the elliptical model [Eq. (3)] with this approximate pattern and use Eq. (4)to estimate the directivity when the E- and H-plane beamwidths are different.