# Energy Consumption Model for Sensor Nodes Based on LoRa and LoRaWAN
## 5 LoRa and LoRaWAN Technologies
### LoRa Modulation Technique
#### 5.2.3 Acknowledgment Transmission, Communication Range and Sensitivity  
As presented in Figure 6, following each uplink transmission, the sensor node opens two short receive slots in the Class A. The first \(RX1\) message uses the same frequency channel as the uplink frame (the RX Delay 1 is equal to 1 s for the _XX1272_ LoRa transceiver). The second receive window \(RX2\) uses a fixed configurable frequency and data rate (the RX Delay 2 is equal to 2 s for the _XX1272_) [30]. These RX messages can be considered as a transmission acknowledgement (ACK). The internal structure of the ACK message is depicted in Figure 6. The ACK packet is ending with the Message Integrity Code (MIC). To calculate the communication range of LoRaWAN system, different statements can be found reaching from multiple kilometers to a maximum range of 15 km [27]. The communication range noted \(d\) can be estimated using the path-loss \(L_{path}\) expression:  
\[L_{path}=(\frac{4\cdot\pi\cdot f}{c})^{2}\cdot d^{n}, \tag{22}\]  
where \(f\) is the LoRa frequency, \(c\) is the light speed and \(n\) is the path-loss exponent, it can be equal to 2 (for free space), 3 (for urban area) and 6 (for high obstruction). Then, the link budget \(L_{budget}\) for the transmission path is expressed as:  
\[L_{budget}=\frac{P_{Tr}}{S_{R}(SF,BW)}, \tag{23}\]  
where \(P_{Tr}\) and \(S_{R}(SF,BW)\) are, respectively, the transmitted power and the receiver sensitivity, which depends on spreading factor and bandwidth. The sensitivity of the receiver is defined by the minimum received power to detect the signal. This sensitivity is obtained for a minimum Signal  
Figure 6: LoRa ACK structure. \begin{table}
\begin{tabular}{c c} \hline \hline
**Transmission Power (dBm)** & **Power Consumption (mW)** \\ \hline
20 & 412.5 \\
17 & 297 \\
13 & 92.4 \\
7 & 95.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: LoRa _XX1272_ characteristics. to Noise Ratio (SNR) equal to \(\frac{E_{bit}}{N_{0}}\), where \(E_{bit}\) is the energy per bit and \(N_{0}\) is the noise power spectral density. Define \(SNR_{0}\) equal to this minimum:  
\[SNR_{0}=\frac{E_{bit}}{N_{0}} \tag{24}\]  
We have \(E_{bit}=S_{r}\cdot T_{bit}\) where \(S_{r}\) is the received power and \(T_{bit}\) is the bit duration. The relation between \(T_{bit}\) and \(T_{chirp}\) is \(T_{bit}=T_{chirp}\cdot 2^{SF}\). Assuming that \(T_{chirp}=\frac{1}{BW}\), Equation (24) can be written:  
\[SNR_{0}=\frac{S_{r}\cdot 2^{SF}}{NF\cdot k\cdot T\cdot BW} \tag{25}\]  
Then, Equation (25) can be rewritten as the following:  
\[S_{r}=\frac{SNR_{0}\cdot N\cdot k\cdot T\cdot BW}{2^{SF}} \tag{26}\]  
Using the datasheet of the _SX1272_ transceiver [29], the receiver sensitivity can be defined in the following equation:  
\[S_{R}(SF,BW)=SNR(SF)\cdot N_{0}=SNR(SF)\cdot NF\cdot k\cdot T\cdot BW, \tag{27}\]  
where _NF_, \(k\), \(T\) and \(SNR(SF)\) are the receiver architecture noise figure, the Kelvin constant, the temperature and the Signal to Noise Ratio, respectively. Comparing Equations (26) and (27), the \(SNR(SF)\) is given in Equation (28):  
\[SNR(SF)=\frac{SNR_{0}}{2^{SF}}, \tag{28}\]  
where \(SNR_{0}\) equals 15 dB for the _SX1272_ transceiver [29]. Then, to have the estimation value of maximum communication range of LoRa link, we assume that there are no antenna gains and set the path loss \(L_{path}\) equal to \(L_{budget}\). Equation (29) depicts the \(L_{path}\) expression:  
\[L_{path}=\frac{P_{Tr}}{S_{R}(SF,BW)}=\frac{P_{Tr}}{SNR(SF)\cdot NF\cdot k\cdot T \cdot BW}=\frac{P_{Tr}\cdot 2^{SF}}{SNR_{0}\cdot NF\cdot k\cdot T\cdot BW} \tag{29}\]  
Using Equations (22), (23), and (27)-(29), the LoRaWAN range noted \(d\) can be estimated as follow:  
\[d=\left(\frac{L_{path}}{(\frac{4\cdot\pi\cdot f}{c})^{2}}\right)^{\frac{1}{d} }=\left((\frac{c}{4\cdot\pi\cdot f})^{2}\cdot\frac{P_{Tr}\cdot 2^{SF}}{ SNR_{0}\cdot NF\cdot k\cdot T\cdot BW}\right)^{\frac{1}{d}} \tag{30}\]  
The LoRaWAN range is an increasing function according to \(SF\) (meaning that we should use high \(SF\) values to reach long LoRaWAN range).