# Optimal Design of Switchable Wearable Antenna Array for Wireless Sensor Networks
## 2 Models and Methods  
### Direct Problem Formulation  
The direct problem can be formulated in the following way: \"given antenna placement and phase shift values calculate the antenna radiation pattern.\" For this purpose, we started our investigation with numerical models of wearable antenna array radiation patterns. We applied the Remcom XFdtd(r) program that utilizes the finite-difference time-domain method (FDTD) [18]. It was used as the reference to elaborate the simplified analytical model of antenna array radiation pattern that was needed in the iterative optimization procedure. The typical approach to computer simulation of wearable antenna radiation pattern uses numerical, heterogeneous models of the human body. In the Remcom XFdtd(r) program, the heterogeneous model of the human body is available with different voxel sizes, from 1 to 10 mm. It represents the entire body, with its internal structure (see Figure 2a). We used this model with 5 mm voxel size for preliminary research on wearable an antenna array [17] and as a reference to the simplified models. In this research, we aimed at optimizing the wearable antenna array by finding the optimum placement of elementary antennas on the body. This requires carrying out numerous simulations of array radiation pattern for different positions of the elementary antennas with respect to the body, preserving the constant distance of antennas to the body surface. In the case of anthropomorphic models, where the body shape is complex, maintaining a fixed distance between the antenna and body is difficult. For this reason, we utilized in Remcom XFDTD software the cylindrical body model presented in Figure 2b to investigate how the radiation pattern of the wearable antenna array depends on its geometry and phasing. This model consists of six cylinders that are made of tissue simulant material, and its usability for wireless body area network simulations was successfully verified in our previous research [19; 20]. The radiation patterns of the wearable antennas obtained with the cylindrical model were in good agreement with the one obtained with the heterogeneous model [21]. The cylindrical model is convenient for application to WBAN optimization when the antenna position is variable, but its distance to the body surface is fixed [22; 23]. In this case, the position of the antenna can be controlled easily with two coordinates \(r_{a}\), \(\psi_{a}\) in a cylindrical coordinate system presented in Figure 2b. For an iterative optimization, the numerical burden of simulation with body model and Remcom(r) XRdtd program is relatively high because the antenna radiation pattern for each combination of phasing is obtained with simulations which take a few minutes each. We propose a simplified, analytical model of a wearable antenna array to overcome this limitation. This approach assumes that the considered wearable antenna array is a planar circular i.e., each antenna is located on a surface of a cylindrical model of the trunk, and its \(z\) coordinate is equal to 1.3 m above the feet. The simplified geometry of the antenna array is presented in Figure 3. This planar circular array is formed by vertical dipoles placed in a fixed distance from the body (\(r_{a}=167\) mm), and each dipole position is defined by \(\psi_{a}\) angle \(G_{a}(\varphi)\). The radiation pattern of a single wearable antenna depends on the antenna position towards the body. The maximum value of antenna gain is directed in the direction that is opposite to the trunk. To model this feature, we proposed an analytical formula to calculate the gain \(G_{a}(\varphi)\) in the horizontal \(x\)-\(y\) plane of the wearable dipole, which simultaneously depends on antenna position angle \(\psi_{a}\) defined in Figure 2b. The directivity pattern of the elementary antenna is modeled by Equation (1), which is a heuristic approximation proposed by the authors.