## 3 SWIPT-Based Multi-Hop Uneven Clustering Routing Protocol Design for RF EH-CRSNs
### Theoretical Derivation of Cluster Radius
#### 3.3.2 Determining the Cluster Radius of the Outermost Ring \(R_{z}\)  
To minimize the collective energy consumption of the outermost ring and the internal rings acting as data relays, the cluster radius value of the outermost ring is calculated. The total energy consumption for the outermost ring comprises the energy expenditures of CHs and CMs, as detailed in Equation (26) and Equation (27), respectively.  
\[\begin{array}{ll}E_{tcon\_CHs(z)}&=E_{conCH(z)}\times N_{CH(z)}=\Big{[}3L_{1} \times\Big{(}E_{elec}+E_{fs}\times R_{z}^{2}\Big{)}+3(N_{z}-1)E_{elec}\times L_ {1}+(N_{z}-1)E_{elec}\times L_{2}\\ &+N_{z}\times E_{DA}\times L_{2}+L_{2}\times\Big{(}E_{elec}+E_{fs}\times d_{CH( z)\to CH(z-1)}^{2}\Big{)}\Big{]}\times\frac{A_{z}}{S_{z}}\end{array} \tag{26}\]  
\[\begin{array}{ll}E_{tcon\_CHs(z)}=&E_{conCM(z)}\times N_{CM(z)}=\Big{[}3L_{1} \times\Big{(}E_{elec}+E_{fs}\times R_{z}^{2}\Big{)}+2N_{z}\,E_{elec}\times L_ {1}\\ &+L_{2}\times\Big{(}E_{elec}+E_{fs}\times d_{CM\to CH(z)}^{2}\Big{)} \Big{]}\times(N_{z}-1)\end{array} \tag{27}\]  
where \(E_{conCH(z)}\) and \(E_{conCM(z)}\) represent the average energy consumption of a single CH and a single CM per round in the outermost ring, respectively; \(N_{CH(z)}\) and \(N_{CM(z)}\) signify the total count of CHs and CMs in the outermost ring, respectively; and \(d_{CM\to CH(z)}\) indicates the average Euclidean distance from CMs to their corresponding CH in the outermost ring.  
The overall energy expenditure of the inner rings, which relay data packets from the outermost ring, is given in Equation (28):  
\[E_{ndd(z)}=\sum_{j=0}^{z-1}\bigg{[}\frac{A_{z}}{S_{z}}\times L_{2}\times\Big{(} 2E_{elec}+E_{fs}\times d_{CH(j)\to CH(j+1)}^{2}\Big{)}\bigg{]}, \tag{28}\]  
where \(CH(0)\) denotes the sink; and \(d_{CH(0)\to CH(1)}^{2}\) represents the average squared distance from the sink to the CHs in the first ring.  
In summary, the total energy consumption of the outmost ring and the inner rings for relaying data packets can be expressed as  
\[E_{total(z)}=E_{tcon\_CHs(z)}+E_{tcon\_CHs(z)}+E_{ndd(z)}=aR_{z}^{4}+bR_{z}^{2} +c\frac{1}{R_{z}^{2}}+e, \tag{29}\]  
where \(a\), \(b\), \(c\), and \(e\) are positive constants, as delineated in Equations (30)-(33). By taking the first-order derivative of Equation (29) with regard to \(R_{z}^{2}\) and setting the derivative to 0, the cluster radius of the outermost ring \(R_{z}\) is ascertained by extracting the square root of the derived result.  
\[a=\frac{1}{R_{t}}\bigg{[}E_{fs}\times(3L_{1}+L_{2})+\frac{2}{R_{t}}\times L_{ 1}\times E_{elec}\bigg{]} \tag{30}\]  
\[b=\frac{E_{elec}}{R_{t}}\times(3L_{1}+L_{2})-\bigg{[}E_{fs}\times\bigg{(}3L_{1} +\frac{L_{2}}{2}\bigg{)}+\frac{2}{R_{t}}\times L_{1}\times E_{elec}\bigg{]} \tag{31}\]  
\[c=\Big{[}(2z-1)\times L_{2}\times R_{t}^{2}\Big{]}\times\bigg{[}\frac{4}{9}z \times E_{fs}\times R_{t}^{2}+2(z-1)\times E_{elec}\bigg{]} \tag{32}\]  
\[e=(2z-1)\times R_{t}\times\Big{[}3L_{1}\times\Big{(}E_{fs}\times R_{t}+E_{elec }\Big{)}+L_{2}\times(E_{DA}+E_{elec})\Big{]}-E_{elec}(3L_{1}+L_{2}) \tag{33}\]