# Specific Resonant Properties of Non-Symmetrical Microwave Antennas
## 2 Materials and Method
### Method for Evaluating Resonant Properties of Microwave Antennas  
From a practical point of view, this article focuses on the frequency dependence of an antenna to a change in its mechanical dimensions. Specifically, it is a matter of determining the resonant frequencies of a set of antennas of certain lengths. Each antenna is in resonance under certain conditions related to its impedance. To explain the innovative view of this article towards this problem, it is necessary to outline a brief theory of the impedance of an antenna \(Z_{A}\). Let's consider the impedance of the antenna at its input terminals \(Z_{IN}\). The input impedance of the antenna generally includes the radiation impedance \(Z_{\Sigma}\) and the loss impedance \(Z_{LOS}\):  
\[Z_{A}=Z_{IN}=Z_{\Sigma}+Z_{LOS} \tag{1}\]  
At frequencies in the UHF band, the antennas are realized as self-supporting and are made of a well-conducting material. Therefore, the losses in this band are negligible, so we can determine \(Z_{LOS}=0\). In general, the impedance is a complex number that has a real \(R\) and an imaginary \(X\) component. Then, the radiation impedance is defined as:  
\[Z_{\Sigma}=R_{\Sigma}\ \pm jX_{\Sigma} \tag{2}\]  
where \(R_{\Sigma}\) is the radiation resistance and \(X_{\Sigma}\) is the reactance, which is the reactive component of the impedance. The radiation resistance takes the power from the source, proportional to the power that the antennas emit. The radiation reactance takes power from the source that is not radiated in reality. The lines of force of the electric and magnetic field originate and disappear, but they do not detach from the antenna. From this point of view, reactance is an undesirable component. The specific value of the impedance depends on the wavelength ratio of the operating frequency \(\lambda\) and the length of the antenna arm \(l\), so we speak of the \(l/\lambda\) ratio. In general, this impedance dependence has a periodic character, and its simplified course is shown in Figure 4. The radiation resistance (blue course) changes periodically with the \(l/\lambda\) ratio, but it always has a positive value. The reactance (red line) also changes periodically with the \(l/\lambda\) ratio, from positive to negative values. With positive values, the antenna has an inductive character (inductance \(X_{L}\)). At negative values, the antenna has a capacitive character (capacitance \(X_{C}\)) [17]. The resonance of the antenna occurs whenever the course of the reactance intersects the zero axis, \(X_{\Sigma}=0\). In this situation, the radiation impedance is proportional only to the radiation resistance, \(Z_{\Sigma}=R_{\Sigma}\), and the antenna emits HF energy with the minimum possible losses. This situation is periodically repeated for each whole multiple of the ratio \(l/\lambda=0.25\). According to the ordinal number of the multiple of the given ratio (1, 2, 3, 4,...), we speak of the first, second, third, fourth, etc., antenna resonance. In antenna practice, only the first and second resonances are used, because unsuitable radiation characteristics of the antenna are formed at other resonances. A more detailed graphical representation of only these two resonances is shown in Figure 5a. Figure 4: The simplified general course of antenna impedance. In Figure 5a, the highlighted course represents the dependence of the thickness of an antenna to its impedance. This course represents the regularity when a thinner dipole (dotted lines) has a more significant change of the value of impedance on the ratio of \(l/\lambda\) and intersects the zero-impedance axis with a steeper slope than the thicker dipole. At the same time, it can be observed in Figure 5 that these two dipoles do not intersect the horizontal axis in one place. The perpendicular vertical axes represent the position of the exact ratio of \(l/\lambda=0.25\); 0.5; 0.75, etc. However, the antenna resonance does not occur exactly at these ratios, but a little earlier. The exact position of this point depends on the slenderness of the antenna \(\Omega\).