# Radar Systems - Phased Arrays

A single antenna can emit a certain amount of power in a particular direction. Obviously, the amount of radiating power will be increased when we use a group of antennas together. The group of antennas is called ** Antenna array **.

An Antenna array is a radiating system comprising radiators and elements. Each of these radiators has its own induction field. The elements are placed so close that each is within the neighbor's induction field. Therefore, the radiation pattern produced by them would be the ** vector sum ** of each.

The antennas radiate individually and while in an array the radiation of all elements is upward, to form the radiation beam, which has high gain, high directivity and better performance, with minimal losses.

Aantenna array is said to be a ** phased antenna array ** if the shape and direction of the radiation pattern depends on the relative phases and amplitudes of the currents present at each antenna of this array.

## Radiation diagram

Consider "n" isotropic radiation elements, which when combined form an ** array **. The figure below will help you understand the same. Let 's the spacing between successive elements of units ' d '.

As shown in the figure, all radiation elements receive the same input signal. Thus, each element produces an equal output voltage of $ sin left (omega t right) $. However, there will be an equal ** phase difference ** $ Psi $ between successive elements. Mathematically, it can be written -

$$ Psi = frac {2 pi d sin theta} {lambda}::::: Equation: 1 $$

Where,

$ theta $ is the angle at which the incoming signal is incident on each radiation element.

Mathematically, we can write the expressions for ** output voltages ** of 'n ' radiation elements individually as

$$ E_1 = sin left [ omega t right] $$

$$ E_2 = sin left [omega t + Psi right] $$

$$ E_3 = sin left [omega t +2 Psi right] $ $

$$. $$

$$. $$

$$. $$

$$ E_n = sin left [omega t + left (N-1 right) Psi right] $$

Where,

$ E_1, E_2, E_3,…, E_n $ are the output voltages of the first, second, third,…, n ^{ th } radiation elements respectively.

$ omega $ is the angular frequency of the signal.

We will get the ** global output voltage ** $ E_a $ from the table by adding the output voltages of each element present in this network, since all these radiation elements areconnected in a linear network. Mathematically, it can be represented by -

$$ E_a = E_1 + E_2 + E_3 +… + E_n::: Equation: 2 $$

** Replace **, the values of $ E_1, E_2, E_3,…, E_n $ in equation 2.

$$ E_a = sin left [omega t right] + sin left [omega t + Psi right ] + left sin [omega t + 2 Psi right] + sin left [omega t + left (n-1 right) Psi right] $$

$$ Rightarrow E_a = sin left [omega t + frac {(n-1) Psi)} {2} right] frac {sin left [frac {n Psi} {2} right]} {sin left [frac {Psi} {2} right]}::::: Equation : 3 $$

In equation 3, there are two terms. From the first term, we can observe that the overall output voltage $ E_a $ is a sine wave having an angular frequency $ omega $. But, it has a phase shift of $ left (n - 1 right) Psi / 2 $. The second term in Equation 3 is an ** amplitude factor **.

The amplitude of the equation3 will be

$$ left | E_a right | = left | frac {left sin [frac {n Psi} {2} right]} {left sin [frac {Psi} {2} right]} right | ::::: Equation: 4 $$

We will obtain the following equation by replacing equation 1 in equation 4.

$$ left | E_a right | = left | frac {left sin [frac {n pi d sin theta} {lambda} right]} {left sin [frac {pi d sin theta} {lambda} right]} right | ::::: Equation: 5 $ $

Equation 5 is called ** field strength model **. The field strength model will have the values of zeros when the numerator of equation 5 is zero

$$ sin left [frac {n pi d sin theta} {lambda} right] = 0 $$

$$ Rightarrow frac {n pi d sin theta} {lambda} = pm m pi $$

$$ Rightarrow nd sin theta = pm m lambda $$

$$ Rightarrow sin theta = pm frac {m lambda} {nd} $$

Where,

$ m $ is an integer and is equal to 1, 2 , 3 rdt so on.

We can find the ** maximum values ** of the field strength model using the L-Hospital rule when the numerator and denominator of equation 5 are equal to zero. We can observe that if the denominator of equation 5 becomes zero, then the numerator of equation 5 also becomes zero.

Now, let's get the condition for which the denominator of equation 5 becomes zero.

$$ sin left [frac {pi d sin theta} {lambda} right] = 0 $$

$$ Rightarrow frac {pi d sin theta} {lambda} = pm p pi $$

$$ Rightarrow d sin theta = pm p lambda $$

$$ Rightarrow sin theta = pm frac {p lambda} {d} $ $

Where,

$ p $ is an integer and it is equal to 0, 1, 2, 3 and so on.

If we consider $ p $ to be zero, then we will get the value of $ sin theta $ as zero. In this case, we will get the maximum value of the pattern of intensifield tee corresponding to the ** main lobe **. We will get the maximum values of the field strength model corresponding to the ** side lobes **, when we consider other values of $ p $.

The direction of the phase network radiation pattern can be oriented to vary the relative phases of the current present at each antenna. This is the ** advantage ** of phased electronic scanning.