# Compact Design of Annular-Microstrip-Fed mmW Antenna Arrays
## 2 Theory and Analysis of Annular Microstrip Feeding Design
### Two-Direction Pitchfork-Shaped Microstrip Antenna Array  
As shown in Figure 11, we propose the microstrip antenna array consists of outer annular microstrip line and two pitchfork-shaped array branches as the second design. Each pitchfork-shaped branch is composed by two linear arrays each with \(1\times 3\) rectangular patches. This design is similar to the basic design, but the difference is that we use pitchfork-shaped feeding mode inside the outer annular microstrip line. Initially, the structure was still designed as four array branches. However, the upper and lower branches will influence the radiation characteristics because of the opposition of phase of the two side feeding units. So, we finally adopt two opposite array branches.  
We set the red point as the feeding point which is at the center of the left-hand outer annular microstrip line in order to feed the two asymmetrical radiating pitchfork-shaped branches. The two pitchfork-shaped patch array branches consist of twelve rectangular patches in total.  
The materials and thicknesses of patches, microstrip line, and the mental ground of bottom layer are the same as normal annular-shaped microstrip antenna array. Besides, in order to meet the requirement of the input impedance value 50 \(\Omega\) at the coaxial probe feed port from the observed antenna terminal, the two pitchfork-shaped branches need to be designed for implementing the input impedance of 75 \(\Omega\) and 150 \(\Omega\), respectively.  
We set the inner diameter of the outer annular microstrip line as \(R_{1}\) = 25.870 mm and the width of the outer annular microstrip line as \(d_{2}\) = 0.110 mm, as shown in Figure 9. Other detailed parameters of the structure have been demonstrated in Table 2. The meanings of these parameters are shown in Figure 9.  
Figure 11: Two-direction pitchfork-shaped microstrip antenna array, (**a**) top view, (**b**) side view, (**c**) the left-side pitchfork-shaped branch (input impedance of 150 \(\Omega\)), (**d**) the right-side pitchfork-shaped branch (input impedance of 75 \(\Omega\)).  
The gain and the reflection coefficient of the microstrip antenna array consists of outer annular microstrip line and two pitchfork-shaped array branches have been shown in Figure 12.  
The maximum gain of two-direction pitchfork-shaped microstrip antenna array is 9.65 dBi at 59.8 GHz and the minimum reflection coefficient of it is \(-25.92\) dB at 59.9 GHz, with the impedance bandwidth of 0.25 GHz. Our designs reach a good impedance matching.  
Moduli and the detailed vector plots of the instantaneous surface current distribution of two-direction pitchfork-shaped microstrip antenna array are shown in Figure 13. Without the side feeding units, the only opposite units are the same in phase which relatively increase the gain and the aperture efficiency of the antenna array compared with the first structure. The asymmetric design will impact the current on the branch with larger input impedance relatively little. This will also mean that antenna cannot obtain uniform radiation on more directions.  
Besides, we show the \(x\)-\(z\) plane, \(y\)-\(z\) plane, and \(x\)-\(y\) plane of the radiation pattern of pitchfork-shaped microstrip antenna array in Figure 14.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Value (mm)** & **Parameter** & **Value (mm)** \\ \hline \(L_{2}\) & 0.780 & \(L_{5}\), \(L_{9}\) & 2.445 \\ \(L_{2},L_{6}\) & 0.860 & \(W_{2}\) & 1.050 \\ \(L_{3},L_{7}\) & 2.490 & \(d_{2}\) & 0.11 \\ \(L_{4}\) & 0.445 & \(d_{3},d_{5}\) & 0.055 \\ \(L_{8}\) & 0.390 & \(d_{4}\) & 0.110 \\ \(R_{1}\) & 25.870 & \(d_{6}\) & 0.220 \\ \(R_{2}\) & 1.243 & \(R_{3}\) & 1.215 \\ \hline \end{tabular}
\end{table}
Table 2: Parameters of two-side pitchfork-shaped branches.  
Figure 12: Simulation results of two-directional pitchfork-shaped microstrip antenna array, (**a**) gain of this antenna array, (**b**) reflection coefficient of this antenna array.