## 2 Design and Geometry of the Circular Minkowski UWB Antenna  
The top view, bottom view, and side view of the proposed antenna are presented in Figure 1a, Figure 1b, and Figure 1c, respectively. The antenna design incorporates a fundamental circular patch configuration with two circular slots located on the left and right edges of the radiating patch, as illustrated in Figure 2a, and features a single feedline. To enhance the antenna's impedance matching, we introduce a Minkowski square on top of the circular radiating patch as shown in Figure 2b. Additionally, we take a further step to enhance impedance matching by adding an additional iteration of the Minkowski square, as shown in Figure 2c. This modification is aimed at optimizing the antenna's impedance matching, gain, and efficiency characteristics. To improve the antenna's bandwidth, we implement a partial ground configuration, as demonstrated in Figure 2d. To further expand the antenna's bandwidth to cover the entire UWB frequency range, a central groove is introduced on the ground plane, as depicted  
Figure 1: Geometry of the proposed antenna: (**a**) top view, (**b**) bottom view, and (**c**) side view. in Figure 2e. Figure 2e provides a visual representation of the antenna's ground plane, highlighting the presence of the centrally positioned groove that has been incorporated for this purpose. The central groove plays a crucial role in improving impedance matching by reducing the capacitive loading on the patch antenna. This occurs because the groove creates an area of low electric field strength at the center of the ground plane, which in turn reduces the amount of current flowing through the patch antenna. As a result, the antenna's impedance is more closely matched with that of the feedline, leading to enhanced efficiency. Here are some additional details about the design:  
* The antenna is made of FR-4 material with a dielectric constant of 4.4. * The patch antenna is 17.6 mm in diameter and 1.6 mm thick. * The feedline has 50 ohms impedance. * The center groove is 2.46 mm wide and 1.23 mm deep. * The antenna is designed to operate at a lower cut-off frequency of 3.1 GHz. Figure 2: Design evolution of the circular Minkowski fractal antenna: (**a**) Ant-1, (**b**) Ant-2, (**c**) Ant-3, (**d**) Ant-4, (**e**) Ant-5, and (**f**) the equivalent circuit of the antenna. The initial antenna design follows the design equations of a planar circular antenna as given below [10; 11; 12; 13; 14; 15]:  
\[f_{l}=\frac{7.2}{\left(\left[2a+\frac{a}{4}+p\right]\cdot k\right)} \tag{1}\]  
\[a=\left[\frac{3.2 GHz}{f_{l}\cdot k}\right]-\frac{4\cdot p}{9} \tag{2}\]  
where \(f_{l}\) = lower cut-off frequency, \(p\) = feedgap, \(a\) = radius of circular patch, and \(k\) = constant. The value of \(k\) is a constant, which is 1.15, the feed gap \(p\) is 0.32, and \(f_{l}\) = 3.1 GHz. Substituting these in the formula gives the radius of the patch as 0.8834 cm. The resonant frequency of the circular patch antenna \(TM_{mmo}^{2}\) can be calculated from the equation given below [16; 17; 18]:  
\[f\tau_{mm0}=\frac{1}{2\pi\sqrt{\mu\epsilon}}\left(X^{\prime}mn\right) \tag{3}\]  
where \(X^{\prime}mn\) represents the zeros of the derivative of the Bessel function \(J_{m}(x)\), which determines the order of the resonant frequency. The dominant mode frequency \(TM_{110}^{Z}\) is calculated by using the following equation [16; 19]:  
\[\left(fr\right)_{110}=\frac{1.8412 Vo}{2\pi a_{e}\sqrt{\epsilon r}} \tag{4}\]  
where \(V_{0}\) is the speed of light and \(a_{e}\) is calculated from Equation (5)  
\[a_{e}=a\sqrt{1+\frac{2h}{\pi a\epsilon r}\left(\ln\left[\frac{\pi a}{2h} \right]+1.7726\right)} \tag{5}\]  
For a circular patch antenna, changing the radius of the patch determines the resonant frequency of the antenna.