# Crescent Microstrip Antenna for LTE-U and 5G Systems
## 5 Dual-Band Microstrip Antenna Designed for LTE-U and 5G Systems  
Before developing an appropriate numerical model of the designed antenna in the simulation environment, it is necessary to perform preliminary calculations of its geometric dimensions based on the parameters of the dielectric substrate and the resonance frequencies of the antenna. These activities are aimed at obtaining a preliminary model of the antenna, which will ensure the compliance of the structure with the assumptions made for it and improve and shorten the achievement of the assumed target in the process of the simulation and optimization of the antenna structure. The main assumption for the designed microstrip antenna operating in 5G and LTE-U systems is the frequency range, which should cover the frequency bands from 3.4 GHz to 3.8 GHz (5G system operating frequencies) and from 5.150 GHz to 5.850 GHz (LTE-U system operating frequencies). In addition to the frequency band, another important requirement for the designed antenna are the dimensions of the antenna, which should not be larger than 60 mm \(\times\) 60 mm, and the omnidirectional radiation pattern with elliptical polarization ensuring the reception of signals with all polarities. The main parameter on which the dimensions of the antenna will depend is its resonant frequency \(f_{r}\) and the relative electric permittivity \(\varepsilon_{r}\) of the dielectric layer of the substrate on which it will be made. The thickness of the substrate directly affects the efficiency and bandwidth of the microstrip antenna. As the thickness of the substrate increases, the antenna operating bandwidth increases and its efficiency decreases. One of the methods of obtaining elliptical polarization for the numerical model of the designed microstrip antenna is to use a patch in the form of a circle. The process of determining the parameters of a microstrip antenna with a circular patch is very similar to that of an antenna with a rectangular patch. In order to determine the diameter of the patch, use the following dependencies [19; 38; 39; 40]:  
\[ \begin{array}{l}2R=\frac{F}{\sqrt{\left\{1+\frac{2h}{\pi\varepsilon_{r}T} \left[ln\left(\frac{\pi\varepsilon_{r}}{2h}\right)+1.7726\right]\right\}}}=12. 88\;\mathrm{mm},\;\mathrm{for}\;f_{r}=3.6\;\mathrm{GHz}\\ 2R=\frac{F}{\sqrt{\left\{1+\frac{2h}{\pi\varepsilon_{r}T}\left[ln\left(\frac{ \pi\varepsilon_{r}}{2h}\right)+1.7726\right]\right\}}}=7.79\;\mathrm{mm},\; \mathrm{for}\;f_{r}=5.5\;\mathrm{GHz}\end{array}  \tag{1}\]  
where \(2R\) is the diameter, \(R\) is the radius, \(h\) is the laminate thickness, and \(\varepsilon_{r}\) is the dielectric constant. \[ \begin{array}{l}F=\frac{1.8412\cdot 10^{\beta}}{f_{r}\sqrt{\varepsilon_{r}}}=16. 46\;\mathrm{mm},\;\mathrm{for}\;f_{r}=3.6\;\mathrm{GHz}\\ F=\frac{1.8412\cdot 10^{\beta}}{f_{r}\sqrt{\varepsilon_{r}}}=10.77\;\mathrm{mm},\; \mathrm{for}\;f_{r}=5.5\;\mathrm{GHz}\end{array}  \tag{2}\]  
Based on the above values, in the next step we determine the effective radius from the relationship [19; 38; 39; 40]:  
\[ \begin{array}{l}2R_{e}=2R\cdot\sqrt{\left\{1+\frac{2h}{\pi\varepsilon_{r}2R} \left[ln\left(\frac{\pi 2R}{2h}\right)+1.7726\right]\right\}}=17.19\;\mathrm{mm},\; \mathrm{for}\;f_{r}=3.6\;\mathrm{GHz}\\ 2R_{e}=2R\cdot\sqrt{\left\{1+\frac{2h}{\pi\varepsilon_{r}2R}\left[ln\left(\frac {\pi 2R}{2h}\right)+1.7726\right]\right\}}=11.56\;\mathrm{mm},\;\mathrm{for}\;f_{r}=5.5 \;\mathrm{GHz}\end{array}  \tag{3}\]The last element of the antenna design is to determine the dimensions of the feed line. The calculation of the dimensions of the microstrip feed line with the characteristic impedance \(Z_{C}=50\)\(\Omega\) begins with the determination of the relationship of auxiliary variables \(a\) and \(b\)[19; 38; 39; 40]:  
\[ \begin{array}{c}a=\frac{Z_{\mathrm{c}}}{60}\sqrt{\frac{\varepsilon_{r}+1}{2}} +\frac{\varepsilon_{r}-1}{\varepsilon_{r}+1}\Big{(}0.23+\frac{0.11}{ \varepsilon_{r}}\Big{)}=1.16\\ b=\frac{60\,m^{2}}{Z_{\mathrm{c}}\sqrt{\varepsilon_{r}}}=7.98\end{array}