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The pyramidal horn antenna is one of the most often used horn antennas.

The antenna is used as a primary feed for reflector antennas as well as standard gain reference antennas in antenna measurements. The pyramidal horn is obtained by flaring all four sides of a rectangular wave guide to form a pyramid-shaped horn with a rectangular aperture. The cross-sectional drawings of a typical pyramidal horn antenna are shown in Fig. 4.12.

In order to apply the field equivalence principle and obtain the far fields, we first need to approximate the field distribution over the aperture. For small flare angles, we assume the aperture fields to have a similar shape as that of the TE₁₀ mode distribution in the exciting wave guide but with a phase variation over the aperture in both x and y directions. To determine the phase variation over the aperture we make several assumptions—(a) it is assumed that the wavefront is cylindrical with its phase center at the inter-section line of the two flared sides as shown in Fig. 4.12, (b) the propagation constant within the horn section is assumed to be the same as the free space propagation constant, and (c) it is assumed that there are no higher order modes generated at the discontinuities at the wave guide to horn junction as well as at the horn aperture. All these assumptions are approximately correct for small flare angles but the deviations may be significant for large flare angles. In the x-z and y-z planes the phase centers are not necessarily at the same position. The radius of curvature of the phase front in the x-z plane is given by r_ox= a/{2 tan(Ψh/2)} and in the y-z plane it is given by r_oy = b/{2 tan(Ψe/2)}, where Ψh and Ψ_e are the flare angles in the H and E planes, respectively. The phase of the field at (x^, 0) with respect to the centre of the aperture is given by the product of the path length difference and the propagation constant, k

δ(x^) = k^r²_ox+ x^2− rox



(4.103)

in the x-z plane and

δ(y^) = k



r_oy² + y^2− roy



(4.104) in the y-z plane. Combining these two, the phase variation over the aperture can be written as

δ(x^, y^) = k^r_ox² + x^2− rox+

r²_oy+ y^2− roy



(4.105)

a b

L

x y y

z

x

e

z r_ox

r_oy

Phase front

a_w b_w

Phase front

h (a)

(b)

Fig. 4.12 Geometry of the pyramidal horn antenna (a) E-plane section (b) H-plane section.

For small flare angles, it is common practice to approximate the phase varia-tion to a quadratic form and write the aperture tangential E and H fields as

Ey = E0cos

πx^ a



e^{−jk[2roxx2 +}

y2 2roy]

(4.106) Hx=−E0

η cos

πx^ a



e^{−jk[2roxx2 +}

y2 2roy]

(4.107)

Applying the field equivalence principle we convert these into equivalent magnetic and electric current sheets in the aperture, M_x =−Ey and J_y = H_x, and the far-fields can be computed using the vector potential approach.

Because of the non-uniform phase over the aperture, the far-field compu-tation is fairly complex and, hence, will not be discussed here. The far-field expressions of a pyramidal horn are somewhat similar to those of an open-ended waveguide with the a and b replaced by the horn aperture dimensions.

Because of the non-uniform phase over the aperture, there is some degra-dation in the directivity, and some changes in the shape of the pattern.

Specifically, the pattern nulls start filling up as the flare angle increases (or the phase variation increases).

For a given set of waveguide dimensions and the length of the horn, L, as we increase the flare angle, the aperture dimension as well as the phase error increases. These two have opposing effects on the directivity. The net effect is that, for a given length of the horn, there is an optimum flare angle for which maximum gain occurs. The optimum flare angle is different in the E and H planes because the amplitude distributions are different. In the H-plane the optimum flare angle is obtained when the maximum phase error (at the edge of the aperture in the H-plane) is equal to 3π/4 and in the E-plane it occurs when the maximum phase error is π/2. The directivity loss factors for these phase errors are 0.8 and 0.77, respectively. Thus, if the horn is designed with optimum phase errors in both planes, we have a directivity expression given by

D = 0.8× 0.77 × 8 π²

4πab

λ²  0.5 ×4πab

λ² (4.108)

The factor (8/π²) occurs due to the cosine amplitude distribution in the x-direction and the remaining factor (4πab/λ²) is the directivity of a uniform distribution over the aperture of dimensions a× b.

In practice, for a given set of wave guide dimensions, it may not be pos-sible to satisfy optimum flare angle condition in both planes simultane-ously. But one can choose one of the flare angles to be optimum and the other will generally be slightly less than the optimum. Generally, the losses in the horn are negligible, and hence we can assume the gain of the horn to be the same as the directivity. In a pyramidal horn design, typically the aperture dimensions are chosen to give a desired gain as per Eqn (4.108), and the length of the horn is minimized using the optimum flare angle criterion. Based on the maximum phase error for optimum directivity, we

can work out the optimum flare angles in the E and H planes as

Ψ_h = 2 tan⁻¹

 3λ 4r_ox

_1/2

(4.109)

Ψ_e= 2 tan⁻¹

 λ 2r_oy

_1/2

(4.110)

For the horn to be realizable, the horn length, L, must satisfy the following equations

L = a− aw

2 tan Ψh

(4.111) L = b− bw

2 tan Ψ_e (4.112)

where a_w× bw are the wave guide dimensions and a× b are the aperture dimensions. These two equations may not be satisfied simultaneously but we select the longer of the two lengths and modify the flare angle in the other equation to satisfy both equations. We select the larger of the two lengths so that the flare angle is optimum in one plane and the phase error in the other plane is less than the optimum. Of course, we can select horn length much longer than the minimum length, in which case the directivity loss factor is less and we get a little more gain than predicted by Eqn (4.108).

For a given horn aperture area, ab, the gain varies from 0.81{4πab/λ²} to 0.5{4πab/λ²}; the factor 0.81 is for zero phase error (L → ∞) and the factor 0.5 is for optimum length.