# A Magnetic-Balanced Inductive Link for the Simultaneous Uplink Data and Power Telemetry
## 3 Modeling of The Magnetic-Balanced Inductive Link Circuit
### Signal-To-Interference Ratio  
Since the external receiver acquires the uplink signal from the coil \(L_{3}\), the 500-kHz frequency component (\(V_{3}(\mathrm{j}\omega_{\mathrm{ul}})\)) of \(V_{3}\) is the received uplink signal, and the 2-MHz frequency component (\(V_{3}(\mathrm{j}\omega_{\mathrm{p}})\)) of \(V_{3}\) is the power carrier interference. \(\omega_{\mathrm{ul}}\) and \(\omega_{\mathrm{p}}\) are the angular frequencies of the uplink carrier and the power carrier, respectively. The SIR can be calculated as shown in Equation (7). Equation (7) could be fully expanded by using the symbolic solution of \(\mathbf{I}\). However, the full expansion of Equation (7) is omitted, as the fully expansion results in a long and tedious expression.  
Figure 4: Equivalent circuit model of the magnetic-balanced inductive link circuit.  
\[SIR = \frac{V_{3}\left(\mathrm{j}\omega_{\mathrm{ul}}\right)}{V_{3}\left( \mathrm{j}\omega_{\mathrm{p}}\right)}=\frac{\mathrm{j}\omega_{\mathrm{ul}}M_{13} I_{1}\left(\mathrm{j}\omega_{\mathrm{ul}}\right)+\mathrm{j}\omega_{\mathrm{ul}}M_{23}I_{2a} \left(\mathrm{j}\omega_{\mathrm{ul}}\right)}{\mathrm{j}\omega_{\mathrm{p}}M_{13 }I_{1}\left(\mathrm{j}\omega_{\mathrm{p}}\right)+\mathrm{j}\omega_{\mathrm{p}} M_{23}I_{2a}\left(\mathrm{j}\omega_{\mathrm{p}}\right)}\cdot\frac{1+\mathrm{j}\omega_{ \mathrm{p}}C_{3}R_{3}-\omega_{\mathrm{p}}{}^{2}C_{3}L_{3}}{1+\mathrm{j}\omega_ {\mathrm{ul}}C_{3}R_{3}-\omega_{\mathrm{ul}}{}^{2}C_{3}L_{3}} \tag{7}\] \[\approx \frac{\mathrm{j}\omega_{\mathrm{ul}}M_{23}I_{2a}\left(\mathrm{j} \omega_{\mathrm{ul}}\right)}{\mathrm{j}\omega_{\mathrm{p}}M_{13}I_{1}\left( \mathrm{j}\omega_{\mathrm{p}}\right)}\cdot\frac{1+\mathrm{j}\omega_{\mathrm{p }}C_{3}R_{3}-\omega_{\mathrm{p}}{}^{2}C_{3}L_{3}}{1+\mathrm{j}\omega_{\mathrm{ ul}}C_{3}R_{3}-\omega_{\mathrm{ul}}{}^{2}C_{3}L_{3}}\] (8) \[= \frac{\mathrm{j}\omega_{\mathrm{ul}}k_{23}\sqrt{L_{2}}I_{2a}\left( \mathrm{j}\omega_{\mathrm{ul}}\right)}{\mathrm{j}\omega_{\mathrm{p}}k_{13} \sqrt{L_{1}}I_{1}\left(\mathrm{j}\omega_{\mathrm{p}}\right)}\cdot\left( \mathrm{j}\left(\frac{\omega_{\mathrm{p}}{}^{2}}{\omega_{\mathrm{ul}}{}^{2}}-1 \right)Q_{3}+\frac{\omega_{\mathrm{p}}}{\omega_{\mathrm{ul}}}\right). \tag{9}\]  
Instead of fully expanding Equation (7), we explore simplifying the SIR equation and then perform a qualitative analysis to guide the optimizations of the coil parameters. As \(L_{1}\) is the transmitter coil of the power carrier and \(L_{2}\) is the transmitter coil of the uplink carrier, \(M_{13}I_{1}(\mathrm{j}\omega_{\mathrm{p}})\) is generally much larger than \(M_{23}I_{2}(\mathrm{j}\omega_{\mathrm{p}})\), and \(M_{23}I_{2}(\mathrm{j}\omega_{\mathrm{ul}})\) is much larger than \(M_{13}I_{1}(\mathrm{j}\omega_{\mathrm{ul}})\). Thus, Equation (7) could be approximately simplified to Equation (8). Since \(L_{3}\) resonates at the uplink carrier frequency (\(f_{\mathrm{ul}}\)) with \(C_{3}\), we could derive \(L_{3}C_{3}=1/\omega_{\mathrm{ul}}{}^{2}\). It is also known that the quality factor of the external data coil can be calculated by \(Q_{3}=\omega_{\mathrm{ul}}L_{3}/R_{3}\), and the mutual inductances (\(M_{13}\) and \(M_{23}\)) can be calculated by \(M_{13}=k_{13}\sqrt{L_{1}L_{3}}\) and \(M_{23}=k_{23}\sqrt{L_{2}L_{3}}\). Therefore, Equation (8) can be rewritten as Equation (9). According to the left part of Equation (9), the uplink SIR could be improved by minimizing the coupling coefficient (\(k_{13}\)) and increasing the coupling coefficient (\(k_{23}\)), as \(k_{13}\) is the denominator and \(k_{23}\) is the numerator. According to the right part of Equation (9), the uplink SIR could be further improved by increasing the quality factor (\(Q_{3}\)).