# A Magnetic-Balanced Inductive Link for the Simultaneous Uplink Data and Power Telemetry
## 3 Modeling of The Magnetic-Balanced Inductive Link Circuit  
### Equivalent Modeling  
The equivalent circuit model of the coupling circuit is shown in Figure 4. The power feeding circuit and the uplink transmitter are simplified as ideal AC voltage sources. \(V_{1}\) is the ideal AC voltage source of the power carrier. The internal resistance of \(V_{1}\) is omitted, because it has negligible direct impact on our inductive link circuit optimization. \(V_{\text{ul}}\) is the equivalent AC voltage source of the uplink carrier. \(R_{\text{isolate}}\) is the equivalent total resistance of \(R_{\text{isolate}+}\) and \(R_{\text{isolate}-}\). The internal resistance of \(V_{\text{ul}}\) is omitted as it is negligible comparing to \(R_{\text{isolate}}\). \(L_{1}\), \(L_{2}\) and \(L_{3}\) are the inductance of the external power coil, the implant coil and the external data coil, respectively. \(R_{1}\), \(R_{2}\) and \(R_{3}\) are the equivalent parasitic resistance of these coils, respectively. \(M_{12}\), \(M_{23}\) and \(M_{13}\) are the corresponding mutual inductance between \(L_{1}\), \(L_{2}\) and \(L_{3}\). \(R_{\text{L}}\) is the equivalent resistance of the power load, and its value is about 2 k\(\Omega\) according to our application.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & \(\mathbf{L}_{1}\) & \(\mathbf{L}_{2}\) & \(\mathbf{L}_{3}\) \\ \hline Coil Wire Type & \begin{tabular}{c} Litz wire with 11 \\ strands of 30 AWG \\ \end{tabular} & \
\begin{tabular}{c} Enameled copper \\ \end{tabular} &
\begin{tabular}{c} Enameled copper \\ \end{tabular} \\ Wire Diameter & 0.8 mm & 0.06 mm & 0.06 mm \\ Coil Radius & 20 mm & 10 mm & 8 mm \\ Coil Thickness & 4.5 mm & 0.12 mm & 0.12 mm \\ Number of Turns & \(N_{1}\) = 30 & \(N_{2}\) = 35 & \(N_{3}\) = 25 \\ Self Resonated Frequency & 7 MHz & 6 MHz & 11 MHz \\ \hline Coil Inductance & \multirow{2}{*}{68.2/63.0 \(\mu\)H} & \multirow{2}{*}{72.9/63.6 \(\mu\)H} & \multirow{2}{*}{25.9/24.9 \(\mu\)H} \\ (2 MHz/500 kHz) & & & \\ \hline Equivalent Series & \multirow{2}{*}{8.5/1.9 \(\Omega\)} & \multirow{2}{*}{25.9/16.2 \(\Omega\)} & \multirow{2}{*}{10.2/7.9 \(\Omega\)} \\ Resistance (2 MHz/500 kHz) & & & \\ \hline Quality Factor & \multirow{2}{*}{101/103} & \multirow{2}{*}{35.4/12.3} & \multirow{2}{*}{31.9/9.9} \\ (2 MHz/500 kHz) & & & \\ \hline \end{tabular} The coil inductance, equivalent series resistance and quality factor are measured by a precision impedance analyzer (E4990A, Keysight, Santa Rosa, CA, USA).  
\end{table}
Table 1: Parameters of the coils.  
By applying the mesh-current approach on the equivalent circuit model, Equations (1)-(4) are obtained, where \(s\) is the Laplacian.  
\[V_{1} =I_{1}\left(sL_{1}+R_{1}+\frac{1}{sC_{1}}\right)+sM_{12}I_{2\text{a} }+sM_{13}I_{3}, \tag{1}\] \[0 =sM_{12}I_{1}+I_{2\text{a}}\left(sL_{2}+R_{2}\right)+\left(I_{2 \text{a}}-I_{2\text{b}}\right)\frac{R_{\text{L}}}{1+sR_{\text{L}}C_{2}}+sM_{23 }I_{3},\] (2) \[V_{\text{ul}} =I_{2\text{b}}R_{\text{isolate}}+\left(I_{2\text{b}}-I_{2\text{a }}\right)\frac{R_{\text{L}}}{1+sR_{\text{L}}C_{2}},\] (3) \[0 =sM_{13}I_{1}+sM_{23}I_{2\text{a}}+I_{3}\left(sL_{3}+R_{3}+\frac{ 1}{sC_{3}}\right). \tag{4}\]  
Equations (1)-(4) can be expressed in matrix form, as seen in Equation (5), and can be further rewritten in a vector form as seen in Equation (6), where \(\mathbf{A}\) is the coefficient matrix of the vector \(\mathbf{I}\). Thus, the symbolic solution of \(\mathbf{I}\) can be derived by leveraging Cramer's rule.  
\[\left(\begin{array}{cccc}sL_{1}+R_{1}+\frac{1}{sC_{1}}&sM_{12}&0&sM_{13}\\ sM_{12}&sL_{2}+R_{2}+\frac{R_{1}}{1+sR_{\text{L}}C_{2}}&-\frac{R_{\text{L}}}{ 1+sR_{\text{L}}C_{2}}&sM_{23}\\ 0&-\frac{R_{\text{L}}}{1+sR_{\text{L}}C_{2}}&R_{\text{isolate}}+\frac{R_{ \text{L}}}{1+sR_{\text{L}}C_{2}}&0\\ sM_{13}&sM_{23}&0&sL_{3}+R_{3}+\frac{1}{sC_{3}}\end{array}\right)\left( \begin{array}{c}I_{1}\\ I_{2\text{a}}\\ I_{2\text{b}}\\ I_{3}\end{array}\right)=\left(\begin{array}{c}V_{1}\\ 0\\ V_{\text{ul}}\\ 0\end{array}\right) \tag{5}\]  
\[\mathbf{A}\mathbf{I}=\mathbf{V}. \tag{6}\]