# Specific Resonant Properties of Non-Symmetrical Microwave Antennas
## 2 Materials and Method
### Method for Evaluating Resonant Properties of Microwave Antennas
The slimmess of the antenna is expressed by the ratio of the length of the antenna arm \(l\) to its diameter \(\phi\) or, in general, to the cross-sectional dimension of the antenna arm \(\Omega=l/\phi\)[18]. From a practical point of view, this means that the antenna is not in resonance when its length is exactly \(\lambda/4\), but when it is slightly shorter. This shorter value is expressed by the antenna shortening factor \(\xi\) and it represents the ratio of the actual resonant length of the dipole (e.g., 0.22) to the ratio \(l/\lambda\). For instance, when \(l/\lambda=0.25\) the factor would be \(\xi=0.88\). In practice, this value is in the range from 0.98 to 0.8, but in general aviation, for some type of antennas, it could be less. For aviation applications, in most cases, only the first resonance is used, because it has the smallest resonant length of one arm of the antenna dipole (unipole). In aviation, there is a natural requirement for minimum aerodynamic drag produced by an antenna mounted on the fuselage. Also, for these reasons, the first resonant frequency of each antenna was evaluated in the experimental measurements. Their resonance points on the axis of the ratio \(l/\lambda\), from the reactant's course of the impedance, were evaluated, as shown in Figure 5b. The radiation resistance value of the measured antenna was not evaluated for the purposes of this article. To theoretically clarify the specific properties of flat microwave antennas, as the ideological goal of this article, it is necessary to know the procedure for determining the slope of the reactance line and determine the position of resonance points R on the \(l/\lambda\) axis around the first resonance [19]. The known position of the two points was used to determine the slope of the reactance line. The first of these points is the generally known position of the value of the radiation impedance of the antenna at the exact length of one of its arms \(l=\lambda/4=0.25\), i.e., [20]:  
\[Z_{\Sigma}=73.1\ +\ \mathrm{j}42.5\ [\Omega] \tag{3}\]  
The position of this point is still the same even when changing the slimmess coefficient of the antenna. We can therefore assume that the course of the reactance of each antenna measured by us passes through this point. The position of the second point in the graph is  
Figure 5: Influence of antenna conductor thickness on its impedance: (**a**) Antenna thickness and its impedance in the area of the 1 st and 2 nd resonance; (**b**) Antenna thickness and position of reactance resonance points. determined from the measured resonant frequency at which the antenna is in resonance and the subsequent calculation of the value of ratio \(l/\lambda\). From the generally known antenna reactance waveforms (in the region of the 1 st resonance), this waveform is almost linear in the range of the above points, as per Figure 5a,b. Since we have defined the position of two points for each measured antenna, we can thus display the lines passing through them, which are an important part of the course of their inductance in the region of the first-quarter wave resonance. Figure 6 shows unmeasured but indicative reactance lines of three antennas of different slimmess. The positions, waveforms, and steepness of the individual reactance lines were determined by the technique described above. According to this developed theory, the following section shows the actual results of the reactance course of the measured antennas. The position of the resonant points of the antennas can theoretically be found in Figure 6 in the range of the \(l/\lambda\) ratio from 0 to 0.25. But, practically, the resonance points can only be in two-fifths of this section, i.e., from 0.15 to 0.25. For clarity, in the following figures of a similar type in this section, the displayed area is that shown in the lower part of Figure 6 highlighted by a thick green line. The individual points of the resonant ratio \(l/\lambda\) on the horizontal axis and the slope of the line present interesting data.