## 3 SWIPT-Based Multi-Hop Uneven Clustering Routing Protocol Design for RF EH-CRSNs
### Theoretical Derivation of Cluster Radius
#### 3.3.1 Derivation of the Relationship between Cluster Radii of Adjacent Rings  
The average energy expenditure of a CH in ring \(i\) consists of three components: energy spent on control overhead, energy consumed for data processing within the cluster, and energy utilized for data relaying between clusters, as specifically shown in Equation (17).  
\[E_{sumCH(i)}=3L_{1}\times\left(E_{elec}+E_{fs}\times R_{i}^{2} \right)+3(N_{i}-1)E_{elec}\times L_{1}+(N_{i}-1)E_{elec}\times L_{2}+N_{i} \times E_{DA}\times L_{2}\] \[+L_{2}\times\left(E_{elec}+E_{fs}\times d_{CH(i)\to CH(i-1)}^{2} \right)+\frac{\frac{i-1}{\eta}\frac{A_{i}}{15}\times\left(E_{elec}+E_{fs} \times d_{CH(i)\to CH(i-1)}^{2}\right)\times L_{2}}{\frac{A_{i}}{5}} \tag{17}\]  
The energy received by each CH in ring \(i\) through intra-cluster SWIPT is the sum of the energy transferred to it by its CMs based on their residual energy, as shown in Equation (18).  
\[E_{harCH(i)}=(N_{i}-1)\times E_{elce}\times L_{2}+N_{i}\times E_{DA}\times L_{2}+ \left(E_{dce}+E_{fs}\times d_{CH(i)\to CH(i-1)}^{2}\right)\times L_{2} \tag{18}\]  
Similarly, the average energy consumption and the energy received through intra-cluster SWIPT for each CH in ring \(i+1\) per round can be deduced. Assuming uniform initial energy levels for all CRSN nodes, to ensure a balanced residual energy among CHs across different rings, it is necessary to calculate the energy variation for each CH per round, that is, the net energy consumption \(E_{netCH(i)}\). From Equations (17) and (18), the \(E_{netCH(i)}\) for an individual CH in ring \(i\) can be ascertained.  
\[E_{netCH(i)}=3\Big{(}E_{elce}+E_{fs}\times R_{i}^{2}\Big{)}\times L_{1}+3(N_{i} -1)\times E_{elce}\times L_{1}+\frac{\sum\limits_{j=i+1}^{z}\frac{A_{j}}{S_{j} }L_{2}\Big{(}2E_{elce}+E_{fs}\times d_{CH(i)\to CH(i-1)}^{2}\Big{)}}{\frac{A_{ j}}{S_{i}}}. \tag{19}\]  
Following a similar procedure as Equation (19), the net energy consumption of a single CH in ring \(i+1\) can be calculated. By equating it to Equation (19), the relationship between the cluster radii of ring \(i\) and its adjacent outer ring \(j\) (\(j=i+1\)) can be determined.  
When \(j=z\), the relationship between the cluster radii of adjacent rings can be derived from equal net energy consumption of individual CHs in adjacent rings, as depicted in Equation (20):  
\[R_{z-1}=\sqrt{\left(3L_{1}m_{1}R_{z}\right)^{2}+3L_{1}m_{1}m_{2}}\Bigg{/} \left(3L_{1}m_{1}+\frac{m_{2}}{R_{z}^{2}}\right), \tag{20}\]  
where  
\[m_{1}=E_{fs}+E_{elce}/50, \tag{21}\]  
\[m_{2}=\frac{\Big{(}18E_{elce}\times L_{2}+4E_{fs}R_{t}^{2}\Big{)}(2z-1)}{9 \times(2z-3)}. \tag{22}\]  
When \(j=z-1\), the relationship between the cluster radii of adjacent rings is as indicated in Equation (23):  
\[3L_{1}m_{1}\Big{(}R_{z-2}^{2}-R_{j}^{2}\Big{)}+\left(\frac{2z-3}{R_{j}^{2}}+ \frac{2z-1}{R_{z}^{2}}\right)m_{3}\times L_{2}\times\frac{R_{z-2}^{2}}{2i-1}- \left(\frac{2z-1}{R_{z}^{2}}\right)m_{3}\times L_{2}\times\frac{R_{j}^{2}}{2i+ 1}=0, \tag{23}\]  
where  
\[m_{3}=2E_{elce}+4E_{fs}R_{t}^{2}/9. \tag{24}\]  
By inserting Equation (20) into Equation (23), the derived outcome is as stated in Equation (25).  
\[R_{z-2}=\sqrt{\frac{3L_{1}m_{1}R_{t}^{2}(3L_{1}m_{1}R_{t}^{2}+m_{2})}{2m_{2}^ {2}+3L_{1}m_{1}m_{2}^{2}+3L_{1}m_{1}R_{t}^{2}}}{3L_{1}m_{1}+\left(\frac{(2i-1) \left(3L_{1}m_{1}+\frac{m_{2}}{R_{z}^{2}}\right)^{2}}{(3L_{1}m_{1}R_{t})^{2}+3 L_{1}m_{1}m_{2}}+\frac{2z-1}{R_{z}^{2}}\right)\times\frac{m_{2}L_{2}}{2i-1}- \left(\frac{2z-1}{R_{z}^{2}}\right)\times\frac{m_{2}L_{2}}{2i+1}}} \tag{25}\]  
Following the same logic, the relationship between the cluster radius of any given ring and that of the outermost ring can be established. Therefore, to determine the cluster radius value for each ring, the initial value \(R_{z}\) must be ascertained.