An array of identical elements all of identical magnitude and each with a progressive phase is referred to as a uniform array.

The array factor is given by,

$AF = 1 + e^{j(kdcosθ + β)}+ e^{2j(kdcosθ + β)} + ... + e^{j(n-1)(kdcosθ + β)}$ $$AF = \sum_{n=1}^N e^{j(n-1)(kdcosθ + β)}$$

which can be written as, $AF = \sum_{n=1}^N e^{j(n-1)Ψ} ............... (4)$

Where, $Ψ = kdcosθ + β$

Therefore, multiplying both sides of above eq (4) by $e^{jΨ}$ , it can be written as

$(AF)e^{jΨ} = e^{jΨ} + e^{2jΨ}+ e^{3jΨ} + ...+ e^{(N-1)jΨ} + e^{NjΨ}$................ (5)

Subtracting eq(4) & eq(5),

$AF(e^{jΨ} - 1) = (-1 + e^{NjΨ}) ............ (6)$

which can also be written as,

$$AF = \left [\frac {e^{NjΨ}-1}{e^{jΨ}-1}\right] = e^{j[(N-1)/2]Ψ} \left [\frac{e^{j(N/2)Ψ} - e^{-j(N/2)Ψ}}{e^{j(1/2)Ψ} - e^{-j(1/2)Ψ}}\right]$$

$$= e^{j[(N-1)/2]Ψ} \left[\frac{sin(\frac{N}{2}Ψ)}{sin(\frac{Ψ}{2})}\right]...............(7)$$

If the reference point is the physical centre of the array, the array factor of the above eq. reduces to,

$$AF = \left[\frac{sin(\frac{N}{2}Ψ)}{sin(\frac{Ψ}{2})}\right]........... (8)$$

For small values of Ψ, the above expression can be approximated by,

$$AF = \left[\frac{sin(\frac{N}{2}Ψ)}{(\frac{Ψ}{2})}\right]........... (9)$$

$$(AF)_n = \frac{N}{2}\left[\frac{sin(\frac{N}{2}Ψ)}{(\frac{Ψ}{2})}\right]........... (10)$$

$$(AF)_n = \left[\frac{sin(\frac{N}{2}Ψ)}{(\frac{N}{2})Ψ}\right]........... (11)$$

To find the nulls of array eq(10) & eq(11) are set equal to zero

For n = N, 2N, 3N, . . ., (9) attains its maximum values because it reduces to a sin(0)/0 form. The values of n determine the order of the nulls (first, second, etc.). For a zero to exist, the argument of the arccosine cannot exceed unity. Thus the number of nulls that can exist will be a function of the element separation d and the phase excitation difference β.

The maximum values of (9) occur when

The array factor of (11) has only one maximum and occurs when m = 0 in (13). That is,

$$θ_m = cos^{-1} \left(\frac{λβ}{2πd}\right) ......... (14)$$

which is the observation angle that makes ψ = 0.

For the array factor of (11), there are secondary maxima (maxima of minor lobes) which occur approximately when the numerator of (11) attains its maximum value. That is,

which can also be written as,

$$θ_x ≃ \frac{π}{2} - sin^{-1}\Bigg\{\frac{λ}{2πd}\left[-β ±\left(\frac{2s + 1}{N}\right)π\right]\Bigg\}, \ \ \ \ \ \ \ \ \ s = 1, 2, 3, .... \ \ \ \ \ \ \ \ \ ..............(16)$$

For large values of d(d ≫λ), it reduces to,

$$θ_x ≃ \frac{π}{2} - \frac{λ}{2πd}\left[-β ±\left(\frac{2s + 1}{N}\right)π\right], \ \ \ \ \ \ \ \ \ s = 1, 2, 3, .... \ \ \ \ \ \ \ \ \ ..................(17)$$