# 3D-printed super-Wideband Spidron Fractal Cube Antenna with Laminated Copper
## 2 Antenna Design  
Figure 1a shows the concept of the Spidron fractal, depicting a planar figure that includes a series of contiguous right triangles which are scaled down and added to the hypotenuse of a larger triangle [16]. Each triangle has the identical angular factor of \(\alpha\), with the down-scaling factor \(\delta\) defined as follows:  
\[\delta=a_{n+1}/a_{n}. \tag{1}\]  
Figure 1b illustrates the self-complementary geometry realized by combining Spidron fractal structures. To meet the self-complementary requirements, the side length of the largest triangle of the Spidron fractal is set to \(S/2\), where the side length of the face is denoted as \(S\) and the common angular factor \(\alpha\) is \(45^{\circ}\).  
The geometry of the proposed 3D-printed SWB Spidron fractal antenna with coordinate systems is illustrated in Figure 2. Each face of the proposed antenna consists of a Spidron fractal structure which is composed of ten isosceles right triangles. Each edge has a length \(S\) of 170 mm considering  
Figure 1: Geometry of (**a**) the Spidron fractal; (**b**) the self-complementary structure with the Spidron fractal.  
the lowest operating frequency. A 3D view of the antenna is given in Figure 2a. As shown in this figure, two Spidron fractal patches on each of four faces along the \(x\)- and \(z\)-directions--referred to as open-state faces--are separated by a small gap (see Figure 2b). Meanwhile, the other two sides along the \(y\)-direction--referred to as short-state faces--are where the Spidron fractal patches are connected to each other through a narrow strip (see Figure 2c). Figure 2d shows an expanded planar view of the proposed antenna. Each Spidron fractal patch is attached onto the 3D frame. The frame is made of the 3D printer filament material of acrylonitrile butadiene styrene, which has a dielectric constant of 3.2 and a loss tangent of 0.02. Copper tape is used to realize the Spidron fractal patches.  
Figure 3 shows the simulated input impedance of the proposed antenna as fed by a lumped port. As can be observed, the real part (\(R_{in}\)) and imaginary part (\(X_{in}\)) of the input impedance are around 145-\(\Omega\) and 0-\(\Omega\), respectively. Hence, it is necessary to design a balun to match the impedance of the SMA connector (with the characteristic impedance of 50-\(\Omega\)) and the input impedance of the antenna. Moreover, the unbalanced mode is excited from the SMA connector, but a balanced mode is needed because the proposed antenna operates as a dipole antenna. Therefore, the balun is required not only for impedance matching but also for mode transforming.  
Figure 2: Geometry of the proposed antenna: (**a**) 3-D view; (**b**) Open-state face; (**c**) Short-state face; (**d**) Expanded view.  
Figure 4a,b show the top and bottom sides of the proposed balun, respectively. According to the exponential function, the widths of the ends of the balun (\(w_{0}\), \(w_{1}\) and \(w_{2}\)) are optimized smoothly to transform the impedance from 145-\(\Omega\) to 50-\(\Omega\) in broadband. The exponential function can be expressed as [13]:  
\[y=ae^{bx}, \tag{2}\]  
where \(a\) and \(b\) are constants determined by the size of the balun. The constants \(a\) and \(b\) are calculated by \(w_{0}\), \(w_{1}\) and \(w_{2}\), respectively. The strip lines on the top and bottom sides of the balun change the mode of excitation from the unbalanced to the balanced mode. In addition, the length \(L\) of the balun is determined according to the lowest operating frequency. The derived optimum parameters are as follows: \(w_{0}\) = 10 mm, \(w_{1}\) = 3.4 mm, \(w_{2}\) = 1 mm and \(L\) = 150 mm. The proposed balun is printed onto a 1.52-mm-thick RF-35 substrate with a dielectric constant of 3.5 and a loss tangent of 0.0018. The balun is attached perpendicularly to the open-state face at the +\(z\)-direction in Figure 2d.