## 2 Equivalent Circuit Model and Orientation Analysis of a 3-Coil Inductive Link  
The 3-coil WPT link is an extension of the conventional 2-coil WPT link. The equivalent circuit of the 3-coil system is shown in Figure 2, where \(L_{1}\), \(L_{2}\) and \(L_{3}\), denote the source, transmitter, and receiver coils respectively. The source \(L_{1}\) is added either to the high-Q transmitter coil \(L_{2}\) or receiver coil \(L_{3}\)[22; 23]. In general, for bio-medical applications, the \(L_{1}\) is added to the \(L_{2}\) to minimize the size and complexity of the implanted unit and to optimize the system even after implantation [24]. The \(L_{1}\) and \(L_{2}\) are strongly coupled to each other and in turn are mutually coupled with the \(L_{3}\). \(C_{1}\), \(C_{2}\) and \(C_{3}\) are the resonant capacitances of \(L_{1}\), \(L_{2}\) and \(L_{3}\) respectively and are placed in series with the coils. \(R_{1}\), \(R_{2}\) and \(R_{3}\) are the self resistances of \(L_{1}\), \(L_{2}\) and \(L_{3}\) respectively. \(R_{S}\), \(R_{L}\), \(P_{in}\) and \(P_{out}\) are the source resistance, load resistance,  
Figure 1: Block diagram of (**a**) 2-coil; (**b**) 3-coil; (**c**) 4-coil WPT systems. input and output power of the system respectively. The Equations (1)-(3) represent the mutual inductance (\(M_{ab}\)), quality factor (\(Q_{ab}\)), and PTE of the system [10; 25]. \[M_{ab}=k_{ab}\sqrt{L_{a}L_{b}}\qquad(0<k<1) \tag{1}\]  
\[Q_{ab}=f_{r}/\Delta_{f} \tag{2}\]  
\[\text{PTE}=|\text{S}_{21}{}^{\prime}|^{2}\times 100(\%) \tag{3}\]  
where \(a\), \(b\in\{1\),2,3\(\}\), \(k_{ab}\) is the coupling coefficient between the coils \(a\) and \(b\), \(L_{a}\) and \(L_{b}\) are the self-inductance of the coils \(a\) and \(b\), \(f_{r}\) and \(\Delta f\) are the resonance frequency and resonance bandwidth of the WPT system and \(S_{21}\) is the linear magnitude of the transmission coefficient. The PTE of the WPT system is influenced by the \(M_{ab}\) and \(Q_{ab}\), which in turn depend on the position and geometry of the external and implantable coils [10; 26]. In this paper, the external and implantable coils have a circular shape and are exposed to alignment conditions, namely perfect alignment, lateral and/or angular misalignments. In perfect alignment, the external and implantable coils remain parallel to each other and their center points are collinear. In lateral misalignment, the external and implantable coils remain parallel to each other, but the center of the implantable coil is displaced by a distance \(\delta\). In angular misalignment, the implantable coil tilts from its ideal position and forms an angle \(\alpha\). In the case where both lateral and angular misalignments are present, the implantable coil endures a shift in the axial position in addition to the rotation. The coordinates of the circular shaped coils are  
\[x_{c}=rcos\varphi;\quad y_{c}=rsin\varphi;\quad z_{c}=0\]  
where \(x_{c}\), \(y_{c}\), and \(z_{c}\) are the cartesian coordinates of circular shaped coil and \(r\) is the radius of the external/implantable coil. The coordinates of the coils at various alignment conditions are presented in Table 1 and Figure 3a. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Alignment Cases** & **External Coil** & **Implantable Coil** \\ \hline Perfect alignment & \((x_{c},y_{c},z_{c})\) & \((x_{c},y_{c},z_{c}+d_{ab})\) \\ \hline Lateral misalignment & \((x_{c},y_{c},z_{c})\) & \((x_{c},y_{c}+\delta,z_{c}+d_{ab})\) \\ \hline Angular misalignment & \((x_{c},y_{c},z_{c})\) & \(\left[\begin{array}{ccc}1&0&0\\ 0&cos\theta&-sin\theta\\ 0&sin\theta&cos\theta\end{array}\right]\left[\begin{array}{c}x_{c}\\ y_{c}\\ z_{c}+d_{ab}\end{array}\right]\) \\ \hline Angular and lateral misalignments & \((x_{c},y_{c},z_{c})\) & \(\left[\begin{array}{ccc}1&0&0\\ 0&cos\theta&-sin\theta\\ 0&sin\theta&cos\theta\end{array}\right]\left[\begin{array}{c}x_{c}\\ y_{c}+\delta\\ z_{c}+d_{ab}\end{array}\right]\) \\ \hline \hline \end{tabular}
\end{table}