or (as shown in Fig. 2.6(b))
Fig. 2.6. Linear array antenna. (a) The origin of the coordinate system is at a unit. (b) The origin of the coordinate system is at the center of two units.
Let a = (a0, a1) be the array element coefficient, then the array factors are given as follows:
Only one phase constant difference exists between the above two expressions, which will not affect the pattern. When θ = 90°, namely in the xoy plane, the above array factor can be written as
The power pattern can be expressed as
Figure 2.7 shows the antenna pattern when a = (a0, a1) = (1, 1), a = (a0, a1) = (1, – 1), a = (a0, a1) = (1, –j) with excitation values of d = 0.25λ, d = 0.5λ, and d = λ, respectively.
The main beam-pointing direction of the pattern changes with the relative phases of the excitation values a0 and a1. When the main beam points to ϕ = 0° or ϕ = 180°, the antenna array is called an end-fire array.
As shown in Fig. 2.7, the main lobe width gradually increases with the main lobe of the pattern moving from ϕ = 90° to ϕ = 0°.
In addition, when d ≥ λ, there will be multiple main lobes in the pattern, which is called the grating lobe as shown in Fig. 2.8.
Consider a two-dimensional array and three half-wave oscillators placed along the z-axis, among which one is at the origin on the x-axis and the other is on the y-axis with a spacing d = λ/2, as shown in Figs. 2.9 and 2.10.
The array element excitation values are a0, a1, and a2, and the corresponding position vectors are d1 =
The array factor of the antenna array is
Fig. 2.7. Patterns of array antenna (Fig. 2.6).
Therefore, the normalized gain of the array is
where g(θ, ϕ) is the pattern function of the half-wave oscillator.
In the xoy plane (θ = 90°), the gain pattern is given as
Fig. 2.8. Antenna patterns when d ≥ λ.
Fig. 2.9. Two-dimensional array.
2.2.2Array antenna freedom
A binary array with two weighting coefficients can maximize the response of the antenna in a desired signal direction or produce a zero in an interference direction by adjusting the weighting coefficients, which is defined as a degree of freedom. When M array elements are used, the degree of freedom of the antenna array is M – 1. This property has important applications in the pattern synthesis of array antennas.
Assume that the radiation pattern of the array is
where
Fig. 2.10. Patterns of the two-dimensional antenna array (Fig. 2.9).
which refers to
In Eq. (2.74), when L ≤ M – 1, the equations have a non-zero solution.
And it also needs to establish a constraint equation when it is required by the pattern to produce a maximum in a certain direction.
This is also a homogeneous linear equation for wm. Therefore, it also requires the degrees of freedom of an array when generating a beam maximum in a certain direction.
In a word, there are M weighted M-ary arrays with (M – 1) degrees of freedom, and at most L1 independent beam maxima and L2 = M – 1 – L1 beam zeros can be achieved.
2.2.3Array antenna pattern synthesis
