a Matlab package for phased array beam shape inspection

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electric field and the scattering direction is constant over the whole common volume (in
our case, it is then equal to the elevation angle ɛ from the test array to the center of
the common volume). We use the bistatic radar equation to estimate the received signal
power P s
P s = P x × λ2 D 1 D 2
(4π) 3 R1 2 × sin 2 (χ)σ e × n e V eff , (93)
R2 2
where the effective scattering volume is
∫
g 1 (P )g 2 (P )
V eff = dV (P )
(r 1 /R 1 ) 2 (r 2 /R 2 ) 2 . (94)
CV
Quantities indexed by “1” refer to the VHF antenna system, quantities indexed by “2”
refer to the test array system. The quantities g 1 and g 2 are the relative gains of the
VHF antenna and the test array in the direction of the point P in the common volume,
so that the absolutely normalized gains in the direction of P are G 1 (P ) = D 1 × g 1 (P )
and G 2 (P ) = D 2 × g 2 (P ). The relative gains are normalized to unity towards the center
of the beams, the point Q.
For the vertical VHF beam, we use a gaussian beam shape approximation in the XY
plane at altitude z
g 1 (x, y) = e − (x/r 1 )2 − (y/r 1 )2
A B . (95)
with A = 3.97 × 10 −5 and B = 3.19 × 10 −5 , which correspond to half power beam width
of 1.7 ◦ in the north-south (= x) direction, and 0.6 ◦ in the east-west (= y) direction, and
r 2 1 = x2 + y 2 + z 2 . Eq. (95) is evaluated by simple vhfbeam.m in the package. We take
the VHF antenna directivity to be D 1 = 43.2 dBi, which corresponds to an effective area
of 3000 m 2 .
The test array directivity D 2 as a function of the zenith angle of the phase-steered
beam in the XZ-plane is computed by plot maxgain.m, which evalutes Eq. (71). For
the array element gain, we use the simple approximation given by Eq. (73), with the
element directivity 11.5 dBi. For the present purpose this seems a fair enough approximation
of the gain diagrams provided by the element’s manufacturer. The directivity
D 2 is shown in Fig. 26. For the zenith angle 35 ◦ (elevation 55 ◦ ), which correspond to
the element Yagi maximum gain direction, and common volume center at 300 km above
the VHF antenna, we find D 2 = 28.1 dBi. The relative gain g 2 of the test antenna is just
taken to be the array factor AF squared, computed from Eq. (50) using arraygain.m.
The presumption is that an element’s beam is wide compared to the arrays’s grating lobe
width. Fig. 28 shows the perpendicular cross section of the resulting test array beam
shape g 2 , projected to the common volume distance R 2 = 360 km, which corresponds to
300 km altitude. As marked in the figure, the half power beam width in the meridian
plane is 3.7 ◦ .
The effective volume Eq. (94) is evaluted by common volume.m by setting up a
sufficiently tight rectangular grid of evaluation points which cover a sufficiently large
3D region around the common volume center point Q, and replacing the integral by the
Riemann sum,
V eff ≈ ∆V × ∑ P
Γ(P ) ≡ ∆V × ∑ P
g 1 (P )g 2 (P )
[r 1 (P )/R 1 ] 2 [r 2 (P )/R 2 ] 2 . (96)
where ∆V = dx dy dz is the volume of the grid’s cell. We take the grid bounding box
∆X × ∆Y × ∆Z to be so large that it fully encompasses at least all the terms Γ(P ) that