# Optimal Design of Switchable Wearable Antenna Array for Wireless Sensor Networks
## 2 Models and Methods
### Direct Problem Formulation
\[G_{a}(\varphi,\psi_{a})=G_{0}cos^{4}\left(\frac{\varphi-\psi_{a}}{2}\right) \tag{1}\]  
where \(\varphi\in[0,\;2\pi)\) is the coordinate in the cylindrical system and \(G_{0}\) is a scaling parameter. The \(G_{0}\) can be used to scale the maximum value of the dipole gain obtained from Equation (1). The proximity of the human body deteriorates the gain of a simple dipole antenna that was used in our research. In this case, the distance of the dipoles to the body model was equal to 10 mm, and the  
Figure 3: Wearable antenna array—simplified geometry of four-element planar circular array. Figure 2: Antenna array on a human body model: (**a**) heterogeneous anthropomorphic model; (**b**) cylindrical model of the human body: \(r_{a}\), \(\psi_{a}\)—coordinates of a single antenna. maximum value of dipole gain in the linear scale was reduced to 0.38. To obtain the best matching of gain calculated with Equation (1) and values obtained from finite difference time domain based (FDTD) simulations with cylindrical model the value of \(G_{0}\) = 0.35 was applied. In Figure 4, the radiation pattern of a single dipole antenna located on body that was obtained with the simplified model given in Equation (1) is compared with results of simulations with FDTD method and cylindrical model of the body for two values of antenna position angle \(\psi_{a}\) equal to \(0^{\circ}\) and \(30^{\circ}\). The results obtained with the use of one and the other model are similar. The angular distribution of antenna gain is similar in both cases. Additionally, the effect of changing the direction of maximum radiation (angle for the maximum value of gain) depending on antenna position \(\psi_{a}\) is preserved in the simplified analytical model. Having the formula that approximates radiation of the single antenna \(G_{a}(\varphi,\psi_{a})\) we calculated the gain of the four-element circular array \(G(\varphi,\psi_{a})\) given by Equation (2) [24]:  
\[G(\varphi,\psi_{a})=\sum_{l=1}^{4}G_{a}(\varphi,\psi_{a}(1))e^{\beta\chi_{a} \cos{(\varphi-\psi_{a}(1))}+\alpha(l))} \tag{2}\]  
where:  
\(\varphi\in[0,\,2\pi)\), \(k\)--wave number, \(l\)--radiator number, \(r_{a}\) = 167 mm--array radius, \(\alpha(l)\)--feeding phase shift of the \(l\)-th antenna, \(\psi_{a}(l)\) angular position of the \(l\)-th antenna. The radiation pattern of antenna array obtained with Equation (2) was compared with the results of simulations with the cylindrical model of the wearable array in FDTD. In Figure 5 the radiation pattern of the four-element antenna array obtained with simulations in the XFdtd program and cylindrical body model are compared to the results obtained with the simplified method. The angular positions of the antennas \(\psi_{a}\) are: \(-20^{\circ}\), \(20^{\circ}\), \(160^{\circ}\), \(200^{\circ}\). The simplified model (Equation (2)) was examined with different values of \(G_{0}\) parameter. The best results were obtained for \(G_{0}\) = 0.16. The angular distributions of antenna gain obtained with FDTD and Equation (2) are similar, but the simulation time was significantly shorter in the case of the simplified model given by Equations (1) and (2). Having the simplified model given in Equation (2), we examined the ability of the proposed four-element wearable array to form its radiation pattern for interference rejection. In this experiment, we assumed the angular positions of the antennas \(\psi_{a}\) are: \(-20^{\circ}\), \(20^{\circ}\), \(160^{\circ}\), \(200^{\circ}\) and the signal comes from direction \(\varphi_{s}\) = \(90^{\circ}\) while the interference is located at \(\varphi_{i}\) = \(0^{\circ}\). For such a case, presented in Figure 6, the array with phasing vector \(\alpha\) = \([0^{\circ},0^{\circ},0^{\circ}]\) direct the main beam with the greatest value of gain towards interference and the minimum value of gain towards the signal.