## 2 A New and Simple Design Method for End-Fire Antenna Array Design  
Figure 1a shows a basic architecture of an N-element end-fire antenna composed of dipole antennas. N stands for the number of antenna units. d is the distance between antenna elements. Usually d = M * \(\lambda\)/4, M = 1,2,3... [1], and M is an integer to describe the distance d. When M = 1, an end-fire antenna array: a helix antenna could be realized such as in ref. [6]. Nevertheless, when M = 2, a new and simple design method for end-fire antenna design is discovered that means that, when antenna elements are \(\lambda\)/2 apart, a simple end-fire antenna array could be designed and constructed easily without matching networks between antenna elements. From Figure 1b, we can see that because the antenna elements are \(\lambda\)/2 apart, RF signals received or sent from neighboring antenna elements are out of phase. For EM signals with a distance of half the wavelength, they strengthen in the direction of the +X axis, while cancelling in the direction of the \(-\)X axis. This the reason why an end-fire antenna can be constructed without matching networks when two dipole antennas are placed with a distance of \(\lambda\)/2. This is why antenna elements could be connected together directly to form an end-fire antenna array without any impedance matching networks. The rule that an end-fire antenna array could be constructed without matching networks when neighboring elements are \(\lambda\)/2 apart could also be proved in a more rigorous analysis, as follows. (1) First of all, let us consider the first end-fire antenna: the two dipole antenna elements are \(\lambda\)/2 apart, and no auxiliary dipole element is inserted between them, as shown in Figure 2a,b. In Figure 2a, at 24 GHz with Rogers 4350 technology, L = d, d = 2.3 mm, M = 2, N = 2. Figure 1: (a) N-element end-fire antenna composed of dipole antennas and (b) minimum RF interferences or best isolation when antenna elements are \(\lambda\)/2 apart. In this circuit, characteristic impedance of transmission line \(Z_{0}\) is almost equal to the load of the dipole antenna \(Z_{\mathrm{L}}\). According to transmission line theory and the theory of elements current for series-fed microstrip arrays [7,8], after a transmission line of \(\lambda/2\), input impedance of \(\mathrm{R_{L}}\) remains the same. According to the theory of the circuit model, the ratio of impedances is as follows: \(Z_{\mathrm{2i}}\):\(Z_{\mathrm{L}}\) = 1:1; therefore, the ratio of the currents of the two dipole antennas can be obtained as follows: \(\mathrm{I_{1}}\):\(\mathrm{I_{2}}\) = 1:1. Consequently pattern function of the array can be obtained as follows:  
\[f(\theta)=\frac{\cos\bigl{(}\frac{\pi}{2}\sin\theta\bigr{)}}{cos\theta}*\left| 2\cos\Bigl{(}\frac{\pi}{2}\bigl{(}1-cos\theta\bigr{)}\Bigr{)}\right| \tag{1}\]  
In Equation (1), \(\frac{\cos\bigl{(}\frac{\pi}{2}\sin\theta\bigr{)}}{cos\theta}\) is the pattern of the dipole antenna, while the remaining part is the array pattern. Additionally, the gain pattern of the two-element end-fire dipole antenna array in the H plane is as shown in Figure 3.  
(2) Secondly, let us consider a second case of an end-fire antenna: the two dipole antenna elements are \(\lambda/2\) apart, and one auxiliary dipole element, \(\mathrm{I_{M3}}\), is inserted between them, as shown in Figure 4. In Figure 4a, at 24 GHz with Rogers 4350 technology, L = d, d = 2.3 mm, M = 2, N = 2. Figure 3: Gain pattern of the two-element end-fire dipole antenna array with no auxiliary dipole antenna in H plane. Figure 2: (**a**) Two dipole antenna elements are \(\lambda/2\) apart with no auxiliary dipole antenna elements; (**b**) the circuit model. According to the theory of the circuit model, the ratio of impedances is as follows: \(Z_{2i}\):\(Z_{3i}\):\(Z_{L}\) = 2:1:1; therefore the ratio of the currents of the three dipole antennas can be obtained as follows: 1:1:3:12 = 4:1:1.