## 2 Hybrid Charging Scheme
### Charging by Mobile Charger with Directional Antenna  
The power Equation (2a) can be considered to be effective between omnidirectional antennas, corresponding to 0 dB antenna gain here. Equation (2a) can be modified to consider the directional antenna of the mobile charger. The directional antenna has a directive gain, which is the ability to concentrate the transmitted energy in a particular direction or orientation [23; 24]. The use of directional antennas has been proven efficient for communications in fading situations. In [25], a directional antenna was considered in a WiMAX communication system attached to the rooftop of a moving car, and demonstrated the advantage of directional antenna in relation to omnidirectional antenna for outage and bit error rate (BER) in a Nakagami-m fading channel. In [26] a directional antenna was utilized for a relay scenario with a Rayleigh fading channel, and it was shown that the outage probability can be improved with the variation of the gain and beamwidth.  
Let \(\phi=0^{\circ}\) be the direction corresponding to the antenna gain and let \(\phi_{j}(t)\) be the orientation angle \(\phi\) of the \(j\)-th sensor node at time \(t\). Owing to the mobility of the charger, the orientation angle of the \(j\)-th infrastructure sensor node is a function of time \(t\). Then, the power equation with directive gain \(g_{A}(\phi_{j}(t))\) in the linear scale, where the antenna type \(A\) denotes the specific antenna gain, is given by  
\[P_{r}^{A}(\overrightarrow{r}_{j},\overrightarrow{r}_{charger}(t),\phi_{j}(t) )=g_{A}(\phi_{j}(t))\frac{\alpha}{\left(d_{j}(t)+\beta\right)^{2}} \tag{6}\]  
Figure 3 presents the radiation patterns of the different types of directional antennas used in this work. The 6 dB antenna gain is obtained by a helical antenna and the 12 dB antenna gain is obtained with a patch array antenna. Note that the directive gain in dB in Figure 3 is relative to the implicit transmit antenna gain in (2a). Helical antennas are known for their high directivity and circular polarization [27], and patch array antennas have many advantages including low cost, light weight, and low profile [28]. However, as our main concern is to investigate the impact of directive gain on charging performance, the details of antenna design are beyond the scope of this paper.  
The directive gain \(g_{A}(\phi_{j}(t))\) in (6) is appropriate for a far-field zone. The typical criterion for separating far and near-field zones is \(d_{j}(t)>2D^{2}/\lambda\)[28], where \(D\) is the maximum dimension of the antenna or array antenna and \(\lambda\) is the wavelength. Considering typical RF frequencies of 1-2 GHz [22] for wireless rechargeable sensor nodes and the small \(D\) (\(\sim\)0.5 m) of helical and patch array antennas designed in this work for such transmit frequencies, the far-field zone starts from several meters of \(d_{j}(t)\). Because the works in [10; 11] did not consider zone separation with power Equation (2a), power Equation (6) is also assumed effective regardless of \(d_{j}(t)\) values. The half-power  
Figure 2: Selection of cluster heads by maximum inclusion criterion: (**a**) inclusion circles, whose centers are at sensor node locations, to obtain cluster heads; and (**b**) sensor nodes in the first cluster are not taken into account for selection of the other cluster heads.  
(3 dB attenuation) beamwidths (\(\equiv\)2B) of antenna gains 6 dB and 12 dB are 100\({}^{\circ}\) and 44\({}^{\circ}\), respectively. These radiation patterns are in the typical shapes of the respective antenna gains. However, the lack of analytic expressions in the radiation patterns in Figure 3 considerably complicates the derivation of the optimal charging path. Therefore, a suboptimal and sequential approach to charge the cluster heads is attempted in this paper.