*3.1.1 The width of the* WP *patch is expressed as follows*

$$\mathcal{W}\_P = \frac{\mathcal{C}}{2f\_0} \sqrt{\frac{\varepsilon\_r + 1}{2}} \tag{1}$$

where C is the speed and ε<sup>r</sup> the dielectric constant of the substrate.

## *3.1.2 The length of the patch is expressed by the formula*

$$L\_p = \frac{C}{2f\_0\sqrt{\varepsilon\_{ref}}} - 2\Delta\_L \tag{2}$$

where the effective dielectric constant is expressed by the formula

$$
\varepsilon\_{\rm ref} = \frac{\varepsilon\_r + \mathbf{1}}{2} + \frac{\varepsilon\_r - \mathbf{1}}{2} \times \left[\frac{\mathbf{1}}{\sqrt{\mathbf{1} + \frac{12h}{W\_r}}}\right] \tag{3}
$$

Where h is the height of the dielectric substrate, the extension of length Δ*<sup>L</sup>* is expressed by the following formula:

$$\Delta\_{L} = 0.412h \left[ \frac{(\varepsilon\_{\rm rff} + 0.3) \left( \frac{W\_{p}}{h} + 0.264 \right)}{(\varepsilon\_{\rm rff} - 0.258) \left( \frac{W\_{p}}{h} + 0.813 \right)} \right] \tag{4}$$

#### *3.1.3 Dimension of the ground plan*

The dimensions of the ground plane are expressed by the following formulas:

$$\mathcal{W}\_{\mathcal{g}} = \mathcal{W}\_{\mathcal{p}} + \mathsf{G}h \tag{5}$$

$$L\_{\rm g} = L\_{\rm p} + \Theta h \tag{6}$$

*Wg*: Ground plane width in mm, *Lg*: Ground plane length in mm.

The feeding technique used for our antennas is the microstrip feeding technique because of its ease of manufacture and its better reliability according to [8].

#### *3.1.4 Adaptation technique*

The choice of the feeding technique chosen therefore imposes a choice of adaptation technique in our work we have chosen the insed feed adaptation technique. This technique involves inserting notches to bring the impedance of the antenna

down to that of the power line [9]. Each notch is equivalent to a parallel admittance Y with a conductance G and a subceptance B. The following formulas on the conductance of a slot, the mutual conductance and the resistance of the antenna are expressed as follows

$$G\_1 = \frac{1}{120\pi^2} \left[ \left[ \frac{\sin\left(\frac{K\_0 W\_p}{2} \cos\theta\right)}{\cos\theta} \right]^2 \sin^3\theta d\theta \tag{7}$$

The mutual conductance is expressed as follows:

$$G\_2 = \frac{1}{120\pi^2} \left[ \left[ \frac{\sin\left(\frac{K\_0 W\_p}{2} \cos\theta\right)}{\cos\theta} \right]^2 \times j\_0 (K\_0 L\_P \sin\theta) \sin^3\theta d\theta \tag{8}$$

Where *j* <sup>0</sup>: represents the Bessel function of order 0. The resistance of the antenna is

$$R\_{\rm in} = \frac{1}{G\_1 + G\_2} \tag{9}$$

After having developed all the theoretical analysis which allows us to find, using mathematical formulas, the various parameters important for the design of the antenna, we will now proceed to the modeling of the antenna.

### *3.1.5 Antenna modeling*

The previous section allowed us to show how to obtain the different antenna dimensions of a rectangular patch antenna. **Table 1** below illustrates these dimensions.

When modeling the antenna several initiatives were taken first with the proposal of several antenna shapes departing from the known classical shape. The **Figures 2** and **3** below represent the different forms of modeled antennas.

The Other forms of antennas modeled in terms of contribution are illustrated below.


**Table 1.** *Antenna dimension.* *Co-Design Block PA (Power Amplifier)-Antenna for 5G Application at 28 GHz Frequency Band DOI: http://dx.doi.org/10.5772/intechopen.98653*

**Figure 2.** *Rectangular patch modeled: a) with dimension, b) without dimension.*

**Figure 3.** *Proposed patch antenna shape.*
