# A Low-Cost Printed Log-Periodic Dipole Array for DVB-T2 Digital TV Applications
## 2 LPDA Design
L\({}_{\text{V1}}\) = 208 mm, L\({}_{\text{V2}}\) = 88 mm, L\({}_{\text{H1}}\) = 64.5 mm, L\({}_{\text{H2}}\) = 225 mm, W\({}_{\text{FEED}}\) = 2.69 mm; (**c**) Transverse section relative to the coaxial cable connection. L\({}_{\text{Cox}}\) = 15 mm, H\({}_{\text{S}}\) = 1.53 mm; (**d**) Front view relative to the coaxial cable connection; (**e**) 3D view. 4. Determine the length of the longest dipole (which covers the lowest part of the frequency band) using CST Studio Suite (as described in [21,28]);
5. Determine the parameters of the N-1 remaining dipoles of the LPDA scaling by \(\tau^{i-1}\) (\(i=2,\ldots,N\)) the parameters of the longest dipole designed in step 4 [7,21,28]. In our design, we choose a relatively low value of \(\tau=0.85\) to limit the number of dipoles necessary to cover the requested frequency band and, hence, the antenna size. Following [PERSON]'s design curves [7], the optimal value of \(\sigma\) should be around 0.15, but this value would result in an excessively large LPDA, which would not fit in the A4 size. Therefore, we have used \(\sigma\) equal to 0.11. With the selected values of \(\tau=0.85\) and \(\sigma=0.11\), the maximum directivity \(D\) predicted by [PERSON] should be around 8 dBi. However, [PERSON]'s curves are applied to a wire LPDA, whereas for a printed one, lower values are expected [4,11,21,28]. In fact, the directivity \(D\) achieved by our designed antenna is about 7 dBi. The operating frequency band of our antenna is 470-860 MHz, with \(\tau=0.85\) and \(\sigma=0.11\). Then, the estimation of the number of elements of the LPDA antenna can be obtained using the expressions given by [PERSON] [7]:  
\[\text{B}=\frac{f_{max}}{f_{min}}=\frac{860}{470}=1.83 \tag{1}\]  
\[\text{B}_{\text{AR}}=1.1+7.7\cdot\left(1-\tau\right)^{2}\cdot cot\alpha=1.58 \tag{2}\]  
\[\alpha=\arctan\left(\frac{1-\tau}{4\sigma}\right)=\arctan\left(\frac{1-0.85}{ 4\cdot 0.16}\right)=19.65^{\circ} \tag{3}\]  
\[\text{N}=1+\frac{\ln\left(\text{B}\cdot\text{B}_{\text{AR}}\right)}{\ln\left( \frac{1}{\tau}\right)}=7.55 \tag{4}\]  
where B\({}_{\text{AR}}\) is the bandwidth of the active region, B is the antenna's operating bandwidth, and \(\alpha\) is the log-periodic antenna aperture angle. The corresponding value N = 7.55 obtained in (4) must be an integer; therefore, we estimate that 7 elements are required to fulfill the requirements of directivity and bandwidth using the standard approach for the design of wire LPDAs. Since the printed LPDA lies on a dielectric substrate, its geometric parameters cannot be calculated in the same way as a standard wire LPDA [7], because their values are strictly related to the substrate on which the radiating dipoles are printed. We have used here a 1.53 mm thick FR4 dielectric substrate, a very low-cost material with a dielectric permittivity of 4.3 and a loss tangent of 0.025. The thickness of the metallization is 35 \(\mu\)m.  
The requested input impedance for receiving antennas in TV applications is equal to 75 \(\Omega\); therefore, the characteristic impedance of the paired strip feeding line must be 75 \(\Omega\), which corresponds to a strip width of W\({}_{\text{FEED}}\) = 2.69 mm [21,28]. At the lowest operating frequency (470 MHz), the active region includes substantially only the longest dipole, which must be designed to resonate at this frequency. Since the resonant resistance of this dipole must be 75 \(\Omega\), both its width and length have been selected through a cut-and-try procedure using CST Studio Suite [21,28]. We obtain L\({}_{\text{DS}}\) = 124 mm and W\({}_{\text{D}}\) = 13.8 mm for a standard printed dipole (Figure 2b). Hence, the standard dipole requires a total size of L\({}_{\text{tot}}\) = 2 \(\times\) L\({}_{\text{DS}}\) + W\({}_{\text{FEED}}\) = 2 \(\times\) 128 mm + 2.69 mm = 250.69 mm, which is significantly larger than the vertical A4 dimension of 210 mm, making it impossible to fit the LPDA, designed with standard printed dipoles as radiating elements, in the A4 size.