# Compact Design of Annular-Microstrip-Fed mmW Antenna Arrays
## 2 Theory and Analysis of Annular Microstrip Feeding Design  
### Outer-Annular Microstrip-Fed Antenna Array  
To achieve the compact feature, the outer annular microstrip line has been analyzed and designed by us. No matter how many branches there are in each structure, we follow the principle that the length of the outer annular microstrip line between every two branches should be the integral multiple of the half-wavelength of the central frequency. For instance, as shown in Figure 1a, the annular microstrip line is divided into four parts. So, the length of each part of annular microstrip line should be designed as above. If this requirement has been met, the quantity of the branches can be designed as 2-6 at designer's option. One point that needs to be assured is that the input impedance of each branch should be equal to 50 \(\Omega\) after being calculated in parallel. Besides, the initial values of the width of the microstrip lines should generally meet the following computational formulas [24]:  
\[m=\frac{Z_{\mathrm{c}}\sqrt{2\big{(}\sqrt{\varepsilon_{\mathrm{r}}+1}\big{)}} }{120}+\frac{\varepsilon_{\mathrm{r}}-1}{\varepsilon_{\mathrm{r}}+1}\bigg{(}0.2258+\frac{0.1208}{\varepsilon_{\mathrm{r}}}\bigg{)} \tag{1}\]  
\[\frac{d_{\mathrm{n}}}{q}=\frac{8e^{m}}{e^{2m}-2} \tag{2}\]  
where \(Z_{\mathrm{c}}\) is the impedance of the microstrip line, \(\varepsilon_{\mathrm{r}}\) is the relative dielectric coefficient, \(d_{n}\) (\(n=0\) or \(n=1\)) is the width of the microstrip line, and \(q\) is the thickness of the microstrip line and \(m\) is the interim parameter. Besides, we choose rectangular patches as the radiation elements in our design. When setting the initial values of the patches, we follow the following computational formulas:  
\[\varepsilon_{\rm e}=\frac{\varepsilon_{\rm r}+1}{2}+\frac{\varepsilon_{\rm r}-1 }{2}\bigg{(}1+12\frac{H}{L_{\rm z}}\bigg{)}^{-\frac{1}{2}} \tag{3}\]  
\[\Delta L=0.412H\frac{(\varepsilon_{\rm e}+0.3)\Big{(}\frac{L_{\rm z}}{H}+0.264 \Big{)}}{(\varepsilon_{\rm e}-0.258)\Big{(}\frac{L_{\rm z}}{H}+0.8\Big{)}} \tag{4}\]  
\[f_{\rm r}=\frac{c}{2(W_{\rm z}+2\Delta L)\sqrt{\varepsilon_{\rm e}}} \tag{5}\]  
where \(\varepsilon_{\rm e}\) is the effective relative dielectric coefficient of the substrate, \(H\) is the thickness of the dielectric substrate, \(\Delta L\) is the length of equivalent radiation gap, \(f_{\rm r}\) is the resonant frequency, \(L_{\rm z}\) is the length of each patch, \(W_{\rm z}\) is the width of each patch, and \(c\) is the speed of light in vacuum. The four antenna structures in this paper are flexible and easy to adjust the bandwidth by changing parameters such as the thickness of the dielectric substrate, which means that our designs have good robustness and can meet the general bandwidth requirements of different application scenes. However, employing the equivalent formulas (1)-(5) are not enough to guide us to design the annular-microstrip-fed mmW antenna arrays we need. Hence, all simulations are using a commercial Method of Moments (MoM) solver (Altair Feko) in this paper, which is one of the famous full-wave numerical methods for 3D electromagnetic field computation. As shown in Figure 1, we propose the outer-annular microstrip-fed antenna array as the basic structure. We use an outer contour annular microstrip line to connect four directional linear arrays with 1 \(\times\) 3 rectangular patches. We set the white point as the feeding point which is at the center of the right-hand outer annular microstrip line. The punched hole with the diameter of \(d\) shown in the Figure 1 is designed for the further experimental soldering with the coaxial probe. All the patches, microstrip line and the ground of bottom layer are applied by silver with the thickness of \(q=0.006\) mm. Besides, the whole antenna array is etched on both surfaces of the two sides of the aluminum oxide dielectric substrate with the relative dielectric coefficient of \(\varepsilon_{\rm r}\) = 9.8 and thickness of \(H\) = 0.12 mm [25].