# Magnetic Communication Using High-Sensitivity Magnetic Field Detectors
## 3 Sensitivity Analysis
### Communication Range for Lossless Transmission Mediums
#### 3.2.1 Derivation of the Coil-to-Coil Transmission Range  
In conventional MI communication systems, the properties of transmitter and receiver coil and the coupling between the coils have an influence on the transmission range. The basis for the calculation  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Meaning** & **Parameter** & **Value** \\ \hline Relative permeability of the core & \(\mu_{\text{r}}\) & 1 \\ Quality factor transmitter & \(Q_{\text{T}}\) & 68 \\ Quality factor receiver & \(Q_{\text{R}}\) & 35 \\ Efficiency transmitter & \(\eta_{\text{T}}\) & 2.4\% \\ Efficiency receiver & \(\eta_{\text{R}}\) & 50\% \\ Bandwidth transmitter & \(\Delta f\) & 1.47 kHz \\ Coil radius & \(r_{\text{T}}\), \(r_{\text{R}}\) & 16 mm \\ Number of turns & \(N\) & 20 \\ Inductance & \(L\) & 14.35 \(\upmu\)H \\ Resonant frequency & \(f_{0}\) & 100 kHz \\ Series resistance of power supply & \(R_{S}\) & 3.2 m\(\Omega\) \\ Load resistance & \(R_{L}\) & 129 m\(\Omega\) \\ Series resistance of coil & \(R_{L1}\), \(R_{L2}\) & 129 m\(\Omega\) @ 100 kHz \\ Transmitting power & \(P_{\text{T}}\) & 43.3 dBm \\ Receiver sensitivity & \(P_{\text{R}}\) & \(-\)42 dBm \\ \hline \hline \end{tabular}
\end{table}
Table 2: Prototype system parameters.  
of the communication characteristics is the Agbinya-Masihour model [16]. The MI link budget can be expressed by the following equation:  
\[d_{\rm cc}^{6}\left(1+\frac{r_{\rm T}^{2}}{d_{\rm cc}^{2}}\right)^{3}=\frac{P_{ \rm T}Q_{\rm T}Q_{\rm R}\eta_{\rm T}\eta_{\rm R}\mu_{\rm T}\mu_{\rm T}\mu_{\rm x }R_{\rm T}^{3}\eta_{\rm R}^{3}\pi^{2}}{P_{\rm R}}, \tag{2}\]  
where \(d_{\rm cc}\) is the transmission range, \(P_{\rm T}\) the transmitter power, \(P_{\rm R}\) the receiver sensitivity, \(Q_{\rm T}=\frac{2\pi f_{0}L}{R_{1L}+R_{\rm S}}\) the transmitter quality factor, \(Q_{\rm R}=\frac{2\pi f_{0}L}{R_{12}+R_{\rm L}}\) the receiver quality factor, \(\eta_{\rm T}=\frac{R_{\rm S}}{R_{\rm S}+R_{1L}}\) the transmitter coil efficiency, \(\eta_{\rm R}=\frac{R_{\rm L}}{R_{\rm L}+R_{12}}\) the receiver coil efficiency, \(\mu_{\rm T}\) the relative permeability of the core material of the transmitter coil, \(\mu_{\rm r,R}\) the relative permeability of the core material of the receiver coil, \(r_{\rm T}\) the transmitter coil radius, and \(r_{\rm R}\) the receiver coil radius. Since distance depends on \(Q_{\rm T}\cdot\eta_{\rm T}\) and \(Q_{\rm R}\cdot\eta_{\rm R}\), where \(Q_{\rm T}\cdot\eta_{\rm T}\) is fixed while \(Q_{\rm R}\cdot\eta_{\rm R}\) is a function of load resistance \(R_{L}\), in Table 2 we have optimized \(R_{L}\) in order to approach maximum distance. Equation (2) can be separated in a term with a significant influence on the determination of the distance and a so-called correction term with a very small effect on \(d_{\rm cc}\):  
\[d_{\rm cc}=d^{\prime}\cdot\Delta d, \tag{3}\]  
where  
\[d^{\prime}=\sqrt[6]{\frac{P_{\rm T}Q_{\rm T}Q_{\rm R}\eta_{\rm T}\eta_{\rm R} \mu_{\rm T}\mu_{\rm x}R_{\rm T}^{3}\eta_{\rm R}^{2}}{P_{\rm R}}}, \tag{4}\]  
and the correction term is  
\[\Delta d=\frac{1}{\sqrt{1+\frac{r_{\rm T}^{2}}{d_{\rm cc}^{2}}}}. \tag{5}\]  
If the communication distance is large compared to the transmitter coil radius, the following approximation holds:  
\[\frac{1}{\sqrt{1+\frac{r_{\rm T}^{2}}{d_{\rm cc}^{2}}}}\approx 1,\;\rm{for} \;r_{\rm T}\ll d_{\rm cc}. \tag{6}\]  
This simplifies the link budget, so that the transmission range \(d_{\rm cc}\) can be approximated as \(d_{\rm cc}\approx d^{\prime}\).