# Design and Development of a Wideband Planar Yagi Antenna Using Tightly Coupled Directive Element
## 2 Printed Yagi Antenna Design Principles  
Antenna design based on elliptical shaped microstrip structures has been emerged as unique and important category among the different shaped because of its dual resonant feature and flexibility in optimization. Design of an elliptical shape microstrip patch antenna has more degree of freedom as the resonant frequency is the function of its major axis and minor axis radii as following [32]:  
\[f_{11}^{e,o}=\frac{15}{\pi\text{e}a}\sqrt{\frac{q_{11}^{e,o}}{e_{r}}} \tag{1}\]  
where, \(f_{11}^{e,o}\) is the dual resonant frequency (in GHz) of \(TM_{11}^{e}\) and \(TM_{11}^{o}\) modes with a and e correspond to physical semi major axis (in cm) and eccentricities, respectively. The eccentricities (\(e=\sqrt{1-\left(b/a\right)^{2}}\)) values depends on aspect ratio (b/a) correspond to, semi minor axis /semi major axis [33]. The approximate values of Mathieu function (\(q_{11}^{e,o}\)) can be determined by following set of equations discussed in [32]. It can be seen that dual resonant modes of an elliptic shape structure can be controlled by varying the aspect ratio (b/a) that can be optimized to exhibit larger bandwidth. In Yagi antennas, induced current exhibits progressive phase shift on the rest of parasitic elements which establishes an array structure suitable for supporting travelling wave. For the circuits printed over a dielectric material, it is essential to present a smooth transitions of induced electric field from the first directive element to the last directive element over the complete band of interest. However, abrupt discontinuity because of sharp edges of a rectangular shape geometry, the electric field lines bends at the edges which results in fringing electric fields. Therefore, to avoid sudden discontinuities along the whole structure, we selected elliptical shape geometry for our design. The results revealed by multiple electromagnetic simulations, we found the antenna's elements with an elliptical shape geometry improves the overall performance especially the bandwidth thus support the travelling wave phenomenon better than its counterpart structure. The general design principles and analytical investigations related to printed Yagi antenna has been extensively discussed in the literature [34; 35; 36]. However, it is important to summarize key salient guidelines required to understand the design and physical operational mechanism of the antenna. The printed Yagi antenna requires a balanced feed to the double sided printed dipole elements. Therefore, a 50 \(\Omega\) microstrip to a parallel strip line transition is required. Typically, printed Yagi structure exhibits, low input impedance and lower front to back ratio. These parameters are significantly controlled, respectively (i) the spacing between the active dipole arms and the ground plane called reflector and (ii) the size (typically the length) of the reflector. By increasing the size of reflector, the backward gain reduces resulting in an improved front-to-back ratio. Similarly, the input impedance increase by increasing the separation between the dipoles and the reflector. However, a significant drop in forward gain is observed for a reflector spacing more than 0.3\(\lambda\). In travelling wave antennas, gain enhancement is the function of longitudinal dimensions thus proportional to the larger dimensions (length) of the antenna. In Yagi antennas gain increases by increasing the number of directive elements. However, practically, there is a limit beyond that additional directive element enhances very little to the gain value because of gradual reduction in induced electric field on the farthest directive elements. For our gain requirements (8 dBi-9 dBi), we found maximum four equally spaced directive elements are suitable resulting in overall antenna length to be 140 mm including the ground plane (reflector).