# Analytical Model for the Duty Cycle in Solar-Based EH-WSN for Environmental Monitoring
## 7 Duty Cycle for Energy Neutral Operation
 \tag{38}\]  
On the other hand, \(\int_{0}^{t}\frac{E_{round}(X)}{T_{rnd}}\,du=\frac{E_{round}(X)}{T_{rnd}}t\). Figure 16 plots the evolution of \(E(t,X)\) during a one-day period. Regarding the energy consumption model, we have used the data shown in Table 3, but with \(T_{rnd}=60\) s and \(DC=40\%\). The load \(\sigma(X)\) has been set to 30. Regarding the energy harvesting model, we have considered \(D_{month}=4.87\) kWh/m\({}^{2}\)/day and \(STDHOURS=12.5\) h (data corresponding to Madrid in September), \(E(0,X)=1000\) Joules (initial energy level), a capacity of 3000 Joules and a solar cell with \(S=36\) cm\({}^{2}\) and \(\eta=11.38\%\). As it can be noticed, \(E(t,X)\) exhibits an oscillating behavior while condition Equation (37a) is satisfied. Next, returning to condition Equation (37b), we can reformulate it as follows after very simple manipulations:  
\[E_{out}(T_{D})=E_{round}(X)\frac{T_{D}}{T_{rnd}} \tag{39}\]  
Note that the quotient \(\frac{T_{D}}{T_{rnd}}\) is nothing else but the number of rounds per day. From Equation (39) we can derive the condition for the duty cycle of node \(X\), which is a parameter contained in \(E_{round}(X)\) through Equation (15). In effect, this equation can be rewritten in the following way, by simply recalling that \(T_{slp}=\frac{T_{l}(100-DC)}{DC}\):  
\[E_{round}(X) = \sum_{i=1}^{CH(X)}(\sigma(c_{i}(X))+1)E_{R,i}(X)+(\sigma(X)+1)E_{T }(X) \tag{40}\] \[+ \left(\left|\frac{T_{rnd}}{T_{l}+\frac{T_{l}(100-DC)}{DC}}\right| -(\sigma(X)+1)\right)(E_{l}+E_{sleep})\]  
In order to simplify the process of obtaining a closed-form expression for \(DC\), it is convenient first to analyze the dependence of \(E_{round}(X)\) on this parameter. Figure 17 shows \(E_{round}(X)\) in terms of \(DC\) for different values of \(\sigma(X)\), \(T_{rnd}=60\) s and the rest of parameters as given in Table 3. As it can be noticed, all curves exhibit a linear trend from relatively small values of \(DC\). Additionally, recall from Section 1 that, whereas in battery-operated sensor networks the primary goal was to keep energy consumption as low as possible, in the case of EH-WSN the focus is on enhancing performance as long as energy neutral operation is preserved. Enhancing performance means enlarging the duty cycle of nodes as much as possible, as demonstrated in Figure 18, which shows the decrease of \(\mathcal{E}[k]\) as \(DC\) increases (note that smaller values of \(\mathcal{E}[k]\) imply higher throughput and lower delays). On the other hand, large values for \(DC\) (on the order of 40-50%) clearly correspond to the linear region of all curves in Figure 17. Thus, we can definitely make use of the linear approximation of \(E_{round}(X)\) in terms of \(DC\) for sufficiently large values of the latter. After performing several manipulations, the resulting expression is as follows:  
\[E_{round}(X)\cong I_{rx}VT_{rnd}\frac{DC}{100}+(\sigma(X)+1)E_{l}^{DAR} \tag{41}\]  
Figure 16: Evolution of \(E(t,X)\) during a one-day period, for \(DC=40\%\). We have validated this approximation for several parameter combination. Figure 19 shows the results of one of the tests, which corresponds to most of data contained in Table 3 and \(\sigma(X)=20\). In all cases, we could observe the same behavior: the straight line approaches very accurately the real function from moderately low values of \(D\)C onwards. Figure 19: Linear approximation to \(E_{round}(X)\) from moderately low values of \(DC\), for \(\sigma(X)=20\). Figure 17: Energy consumption per round in terms of the duty cycle. Figure 18: Expected number of tries in terms of the duty cycle.