# A Low-Cost Printed Log-Periodic Dipole Array for DVB-T2 Digital TV Applications
## 2 LPDA Design
The printed dipole has been therefore adequately shaped by meandering its length, obtaining the layout shown in Figure 2a. The width chosen is equal to the standard dipole, \(W_{\rm D}=13.8\) mm, and the dipole shape has been modified to resonate at 470 MHz with a vertical size fitting the A4 dimension of 210 mm and a resistance of 75 \(\Omega\). The resulting meandered dipole has a layout with a vertical size of each arm equal to 97.5 mm, whereas all the other dimensions are reported in the caption of Figure 2. The overall dipole size, considering the distance between its arms (equal to the paired strip width \(W_{\rm FEED}\)) is 197.69 mm, leaving about 6 mm from the vertical borders of the dielectric substrate. The selected shape for the meandered dipole shown in Figure 2a presents, for each arm, one vertical meander (the path \(W_{\rm D2}\)-\(L_{\rm D7}\)-\(W_{\rm D2}\) in Figure 2a) and a U-turn (the path \(L_{\rm D3}\)-\(L_{\rm D2}\)-\(L_{\rm D1}\) in Figure 2a), whose arm \(L_{\rm D1}\) can be chosen to accurately tune the resonant frequency of the dipole. The horizontal depth of the vertical meander \(W_{\rm D2}\) must be chosen to achieve an adequate reduction in the vertical size of the dipole. However, a compromise should be made since a too-large value of \(W_{\rm D2}\) will produce a degradation of the dipole matching and radiation pattern and a stronger coupling between the adjacent dipoles of the LPDA. The reflection coefficients of the two dipoles are shown in Figure 3. The frequency response is very similar, but the meandered dipole has a narrower \(-10\) dB bandwidth due to the electric and magnetic energy stored in the meanders, which increase the antenna Q factor, and hence decrease its percentage bandwidth. In addition, the far-field pattern at the resonant frequency of 470 MHz, shown in Figure 4, is substantially equivalent for the meandered and standard dipoles, with a broadside gain equal to 1.85 dBi and 2 dBi, and a 3 dB beamwidth equal to 84.5\({}^{\circ}\) and 79.5\({}^{\circ}\), respectively. Once the length \(L_{1}=L_{DS}\) and width \(W_{1}=W_{D}\) of the longest printed dipole have been chosen, the lengths and widths of the other dipoles composing the LPDA are computed by using the following expressions [7; 21; 28]:  
\[\frac{L_{n+1}}{L_{n}}=\frac{W_{n+1}}{W_{n}}=\frac{S_{n+1}}{S_{n}}=\tau\qquad n= 1,\ldots,N-1 \tag{5}\]  
With \(\sigma=S_{n}/4L_{n}\), and \(W_{1}=W_{D}=13.8\) mm in our case, the minimum spacing required for accommodating \(N=7\) standard printed dipoles (where \(N=7\) is the estimated number of elements computed in (4) necessary to fulfill the requirements on directivity and bandwidth) is equal to:  
\[ \begin{array}{l}L_{total_{size}}=W_{1}+\sum_{n=2}^{N}(W_{n}+S_{n-1})=W_{1}+ \sum_{n=2}^{N}\left(\tau^{n-1}W_{1}+\tau^{n-2}S_{1}\right)\\ =13.8+\sum_{n=2}^{7}\left(0.85^{n-1}\times 13.8+0.85^{n-2}\times 4\times 0.15 \times 124\right)=\stackrel{{}}{{330\text{ mm}}}\end{array}  \tag{6}\]  
Hence, a printed LPDA using standard dipoles will exceed both the vertical and horizontal dimensions of an A4 sheet. However, using meandered dipoles as radiating elements will also reduce the horizontal size of the LPDA, allowing the array to fit in an  
Figure 4: Far-field pattern for (**a**) the meandered dipole and (**b**) the standard dipole reported in Figure 2.  
Figure 3: Reflection coefficient for the standard dipole and the meandered dipole reported in Figure 2.  
A4 size. Looking at the layout of the proposed printed LPDA in Figure 1, the distance \(S_{n}\) between two adjacent dipoles is computed from their medium point, as indicated, and it must be obtained using the same formula as standard dipoles, i.e., \(S_{n}=4\sigma L_{n}\), where \(L_{n}\) is the vertical length of the \(n\)th dipole. Since the dipole is me meandered, its vertical length, in this case, is significantly smaller if compared to the standard dipole (97.5 mm vs. 124 mm), and this results in a smaller inter-distance between the dipoles and a smaller size of the whole LPDA, which can be computed as  
\[