# Efficient Wireless Charger Deployment for Wireless Reshapeable Sensor Networks
## 5 Analysis
### Grid Point Separation Analysis  
In this subsection, we analyze the relationship of \(S\), \(D\) and \(H\), which are the grid point separation, the effective charging distance and high of the deployment plane, respectively. For an arbitrary sensor node located at position \(v\) = (\(X\), \(Y\), \(Z\)), we can construct a sphere \(O\) with radius \(D\) (i.e., the charging distance). The intersection of \(O\) and the charger deployment plan (i.e., the grid plane) \(P\) is a circle \(C\) with radius \(R=\sqrt{D^{2}-\left(H-Z\right)^{2}}\) centered at \(u\) (Figure 10). For any point \(u^{{}^{\prime}}\) within the area of \(C\), we have \(\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{ \overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{ \overline{   }}}}}}}}}}}}}}}}\)\(D\). So, if there is at least one grid point \(g\) within \(C\), the sensor node located at \(v\) can be covered by a charger deployed at \(g\). When \(Z\) = 0, \(v\) is on the floor plane and the circle \(C\) has the smallest radius \(r=\sqrt{D^{2}-H^{2}}\). As shown in Figure 11, if \(S>\sqrt{2}\ r\), then there may be no grid point within the area of circle \(C\). On the contrary, if \(S\leq\sqrt{2}\ r\)=\(\sqrt{2(D^{2}-H^{2})}\), there is at least one grid point within the area of circle \(C\). In summary, In summary, \(S\leq\sqrt{2(D^{2}-H^{2})}\) ensures that every sensor node is covered by at least one charger deployed at one of the grid points. Below we show that if \(S\ \leq\ r=\sqrt{D^{2}-H^{2}}\), then there is at least two grid points within the area of circle \(C\) centered at \(u\) with radius \(r\). \(S\ \leq\ r\) implies \(S\ \leq\ \sqrt{2}\ r\), which in turns implies there is at least one grid point \(g\) within \(C\) by the fact just mentioned in the last paragraph. Let \(C^{{}^{\prime}}\) be the circle centered at \(g\) with radius \(r\) and let \(a\) and \(b\) be the intersection of \(C\) and \(C^{{}^{\prime}}\) (Figure 12). Since \(C\) and \(C^{{}^{\prime}}\) have the same radius, we have that the angel \(\angle\)_agb_ associated with the minor arc \(\widehat{ab}\) should be larger than 120\({}^{\circ}\); otherwise, \(g\) will not be within \(C\). For neighboring grid points \(g\), \(g^{\prime}\) and \(g^{\prime\prime}\) that are apart from each other with separation \(S\), their associated angle \(\angle\)_\(g^{\prime}\)_gg\({}^{\prime\prime}\)_ is 90\({}^{\circ}\). We have that either \(g^{\prime}\) or \(g^{\prime\prime}\) is within the sector associated with the minor arc \(\widehat{ab}\). Therefore, there are at least two grid points within the area of  
Figure 11: Illustration of no grid point within the area of circle \(C\) with radius \(r\). Figure 10: Illustration of the intersection of the grid plane with an upper half sphere of radius \(D\) (the charging distance) centered at a sensor node. \(C\). That is to say, there are at least two chargers covering a sensor node. In summary, \(S\leq\sqrt{(D^{2}-H^{2})}\) ensures that every sensor node is covered by at least two chargers deployed at two of the grid points. For even smaller separation, we can use the \"dies per wafer\" equation in [25] to estimate approximately the number of grid points within circle \(C\) with radius \(r=\sqrt{D^{2}-H^{2}}\). The equation is shown below:  
\[\text{Dies per wafer}=\frac{\pi\times\left(\text{Wafer Radius}\right)^{2}}{ \text{Die area}}\ -\ \frac{2\pi\times\left(\text{Wafer Radius}\right)}{\sqrt{2\times\left(\text{ Die area}\right)}} \tag{4}\]  
If we take the grid square as the die area and the wafer radius as \(r\), the radius of circle \(C\), then we have the following equation for \(N_{\text{GS}}\), the number of grid squares in circle \(C\):  
\[N_{\text{GS}}=\frac{\pi\times r^{2}}{S^{2}}\ -\ \frac{2\pi\times r}{S\sqrt{2}} \tag{5}\]  
One grid square has four grid points. However, adjacent grid square share grid points. When we count the number of grid points, a grid square is associated with two non-redundant grid points.