# A Magnetic-Balanced Inductive Link for the Simultaneous Uplink Data and Power Telemetry
## Appendix A Modeling of Self- and Mutual Inductance  
The self- and mutual inductances of the coils need to be carefully modeled for the coil parameter optimizations. Figure 10(a) shows the model for calculating the mutual inductance between two circular filamentary coils with a single loop (Coil #1 and Coil #2). The word filamentary means that the coils satisfy the condition \(\frac{r_{\mathrm{w}}}{r}\ll 1\), where \(r_{\mathrm{w}}\) is the radius of the wire and \(r\) is the radius of the coil. Figure 10(b) shows the model for calculating the self-inductance of a circular filamentary coil with a single loop. According to [PERSON]'s law of induction, we have Equations (A1) and (A2), where \(\Psi\) is the magnetic flux linkage of the coil induced by itself and \(\Psi_{12}\) is the magnetic flux linkage of Coil #2 induced by Coil #1. As seen, the inductance can be calculated through dividing the magnetic flux linkage by the current. In the physics definition, the magnetic flux linkage is an integration of the magnetic field (\(\vec{B}\)) covering a surface, and \(\vec{B}\) can be derived by using the Biot-Savart law. \[V=\frac{\mathrm{d}\Psi}{\mathrm{d}t}=L\frac{\mathrm{d}I}{\mathrm{d}t} \Rightarrow\Psi=LI,\] (A1)  
\[V_{2}=\frac{\mathrm{d}\Psi_{12}}{\mathrm{d}t}=M_{12}\frac{\mathrm{d}I_{2}}{ \mathrm{d}t}\Rightarrow\Psi_{12}=M_{12}I_{2}.\] (A2)  
For the case of Figure 10(a), the magnetic field \(\vec{B}\) at each point \(P\) is integrated covering the entire surface of Coil #2 to calculate the magnetic flux linkage through Coil #2. As the magnetic field perpendicular to the surface contributes to the magnetic flux, the magnetic field \(\vec{B}_{\mathrm{z}}\) in the Z axial direction is integrated. The brief calculation procedure of the magnetic flux linkage (\(\Psi_{12}\)) is listed in Equations (A3)-(A8). Then, the mutual inductance \(M_{12}\) between Coil #1 and Coil #2 is derived in Equation (A9). For the coil with multiple turns, the total self-inductance can be calculated as Equation (A14), where \(\delta_{\left(a_{i},a_{j}\right),\left(b_{i},b_{j}\right)}=1\) if \(a_{i}=a_{j}\) and \(b_{i}=b_{j}\). The total self-inductance consists of two parts, which are the self-inductance of loops and the mutual inductance between loops. Note that if two loops are coaxial in the same plane (\(a_{i}\
eq a_{j}\) and \(b_{i}=b_{j}\)), the equations of mutual inductance (Equations (A10) and (A11)) have a poor calculation accuracy in MATLAB due to the divergence of their integrals. By reviewing the formula derivation procedures of mutual inductance and self-inductance for a single loop, we realize that the magnetic flux integration process of the self-inductance is the same as that of the mutual inductance between loops in the same plane. If the layer distance between the two loops is regarded as the wire radius of the single loop, the equation of mutual inductance (Equation (A9)) and the equation of self-inductance (Equation (A12)) can be equivalent. The equation of self-inductance has better integral convergence than that of mutual inductance on the condition of \(d=0\) mm. Thus, it is suggested to use the equation of self-inductance to calculate the mutual inductance when the two loops are coaxial in the same plane, as shown in Equation (A15).