## 2 Designing the \(4\times 1\) Microstrip Square Patch Array Antenna
### The Single Microstrip Square Patch Antenna
#### 2.2.1 Uniform Power Distribution  
Figure 2 shows the geometry and scattering parameter characteristics of the 4 \(\times\) 1 shunt-connected series feed network using a uniform power distribution. Port 1 is connected by a 50 \(\Omega\) subminiature version A (SMA) connector for measurement, and the characteristic impedance, length, and width of the microstrip line connected to port 1 are  
Figure 1: Design and performance of the single microstrip patch antenna: (**a**) the geometry, (**b**) the input reflection coefficient, (**c**) realized gain in the +\(z\)-axis direction, and (**d**) radiation patterns on the y-z and z-x planes at 24.125 GHz.  
\(Z_{\rm c1}=50\)\(\Omega\), \(I_{\rm 1}=1.87\) mm, and \(w_{\rm 1}=0.54\) mm, respectively. Port 1 feeds ports 2 to 5 where four inset-fed square patches are connected. The input impedance of ports 2 to 5 was set to 100 \(\Omega\), with the characteristic impedance and width of the microstrip line connected to ports 2 to 5 at \(Z_{\rm c2}=100\)\(\Omega\) and \(w_{\rm 2}=0.125\) mm, respectively. The spacing among ports 2 to 5 was set to \(l_{\rm 2}=7.49\) mm, which is about one guided wavelength of the microstrip line at 24.125 GHz. If the power loss in the microstrip line is ignored, the input power to port 1 is the sum of the power delivered to ports 2 to 5:  
\[P_{\rm 1}=P_{\rm 2}+P_{\rm 3}+P_{\rm 4}+P_{\rm 5} \tag{5}\]  
where \(P_{\rm 1}\), \(P_{\rm 2}\), \(P_{\rm 3}\), \(P_{\rm 4}\), and \(P_{\rm 5}\) denote the power delivered to ports 1, 2, 3, 4, and 5, respectively.  
Port 1 is located between ports 3 and 4, and two 100 \(\Omega\) microstrip lines branch out to the left and right symmetrically at the end of the 50 \(\Omega\) microstrip line connected to port 1. Two T-junction structures connect the two branched microstrip lines to ports 2 to 5. A quarter-wavelength transformer (QWT) is used at the input microstrip line of the T-junction structure for impedance matching. For instance, since the characteristic impedance of the microstrip lines to ports 4 and 5 is \(Z_{\rm c2}=100\)\(\Omega\), the impedance seen at the junction of the microstrip lines toward ports 4 and 5 is \(Z_{\rm 11}=Z_{\rm c2}/Z_{\rm c2}=50\)\(\Omega\), which is the parallel impedance of the characteristic impedance of the microstrip lines to ports 4 and 5. Therefore, the impedance of the QWT can be calculated as follows:  
\[Z_{\rm c3}=\sqrt{Z_{\rm c2}\times Z_{\rm il}}=\sqrt{100\times 50}=70.71\ \Omega \tag{6}\]  
The corresponding length and width of the QWT are \(l_{\rm 1}=(1/4\times l_{\rm 2})=1.87\) mm and \(w_{\rm 3}=0.29\) mm, respectively.  
For uniform power distribution, the input power at port 1 is delivered equally to ports 2 to 5. Therefore, the power distribution among ports 2 to 5 is 1:1:1:1, and the power delivered to ports 2 to 5 is one quarter of \(P_{\rm 1}\) or \(P_{\rm 2}=P_{\rm 3}=P_{\rm 4}=P_{\rm 5}=(1/4\times P_{\rm 1})\). Since the power at  
Figure 2: Geometry and scattering parameter characteristics of the \(4\times 1\) shunt-connected series feed network using a uniform power distribution: (**a**) the geometry and (**b**) scattering parameters.  
a port is proportional to the square of the voltage, and scattering parameters are defined as the input-output voltage ratio between ports, the scattering parameters between port 1 (input) and ports 2 to 5 (output) should theoretically be \(S_{21}=S_{31}=S_{41}=S_{51}=1/2=-6\) dB. The 4 \(\times\) 1 shunt-connected series feed network with uniform power distribution in Figure 2a was simulated using CST Studio Suite, and the simulated scattering parameters between port 1 (input) and ports 2 to 5 (output) at 24.125 GHz were \(S_{21}=S_{51}=-6.13\) dB, \(S_{31}=S_{41}=-5.96\) dB, respectively. The simulated scattering parameters at 24.125 GHz were close to \(-6\) dB, and this validates the uniform power delivery from port 1 to ports 2 to 5.