## 2 Main Blocks for Wirelessly-Powered Cage System
### Coil Design and Optimization  
#### 2.1.1 Coil Optimization for Conventional Two-/Three-/Four-Coil Inductive Links  
IMDs typically have a tight limitation in their size depending on the applications, resulting in the diameter limitation of Rx coil in the body. In contrast, the Tx coil in the wirelessly-powered cage has more size relaxation in its design. In the inductive link, the coil geometries should be carefully designed to achieve the efficient inductive coupling considering the load and coil separation. Figure 3 shows the physical and electrical configurations of a two-coil inductive link with geometrical parameters used for primary (\(L_{1}\)) and secondary (\(L_{2}\)) coils, where \(d_{\rm in}\) is inner diameter of each coil, \(d_{\rm out}\) is outer diameter of coils, \(N\) is number of turns, and \(z\) is the coil separation between \(L_{1}\) and \(L_{2}\) coils. The mutual inductance between Tx and Rx coils (\(M_{12}\)) is defined by \(M_{12}=k\sqrt{L_{1}L_{2}}\), where \(k\) is the coupling coefficient of the two-coil link. \(M_{12}\) shows the ratio of magnetic flux common to both \(L_{1}\) and \(L_{2}\). \(R_{1}\) and \(R_{2}\) are the series resistance of the Tx and Rx coils, respectively. The quality factor (\(Q\)) of each coil is defined by \(Q=\omega L/R\), where \(\omega=2\pi f_{0}\) and \(f_{0}\) is the power carrier frequency. Although the two-coil inductive link has its optimized solution for highest PTE depending on a given set of \(Q_{1}\), \(Q_{2}\), and \(k_{12}\) based on [8], the optimized load resistance, \(R_{\rm L,PTE}\), is sometimes far from the nominal target load resistance, \(R_{\rm L}\), which is predefined in the IMD application. Therefore, the multi-coil solution such as three- or four-coil inductive links has been widely studied to provide the designer with more degrees of freedom to convert \(R_{\rm L}\) to \(R_{\rm L,PTE}\) and thereby maximize PTE, while they have a potential negative impact on the size-constrained applications. Figure 4 shows the two-, three-, and four-coil inductive links with the lumped circuit model. The PTE of two-, three-, and four-coil inductive links can be calculated based on the basic circuit theory found in [9]. \[\eta_{\rm 2-coil}=\eta_{23}=\frac{k_{23}^{2}Q_{2}Q_{\rm 3L}}{1+k_{23}^{2}Q_{ 2}Q_{\rm 3L}}\cdot\frac{Q_{\rm 3L}}{Q_{\rm L}}, \tag{1}\]  
\[\eta_{\rm 3-coil}=\eta_{23}\eta_{34}=\frac{\left(k_{23}^{2}Q_{2}Q_{3}\right) \left(k_{34}^{2}Q_{3}Q_{\rm 4L}\right)}{\left[\left(1+k_{23}^{2}Q_{2}Q_{3}+k_{34}^{2}Q_{ \rm 3}Q_{\rm 4L}\right)\left(1+k_{34}^{2}Q_{3}Q_{\rm 4L}\right)\right]} \cdot\frac{Q_{\rm 4L}}{Q_{\rm L}}, \tag{2}\]  
\[\begin{array}{l}\eta_{\rm 4-coil}=\eta_{12}\cdot\eta_{23}\cdot\eta_{34}\\ =\frac{\left(k_{12}^{2}Q_{1}Q_{2}\right)\left(k_{23}^{2}Q_{2}Q_{3}\right) \left(k_{34}^{2}Q_{3}Q_{\rm 4L}\right)}{\left[\left(1+k_{12}^{2}Q_{1}Q_{2}\right) \left(1+k_{34}^{2}Q_{3}Q_{\rm 4L}\right)+k_{23}^{2}Q_{2}Q_{3}\right]\cdot\left[1 additional degree of freedom (\(k_{23}\)) to adjust the reflected load onto \(L_{2}\) to be the optimal value, \(R_{\text{L,PTE}}\), compared to the two-coil link. The PTE of the three-coil inductive link is related with \(k_{23}\), \(k_{34}\), \(Q_{2}\), \(Q_{3}\), and \(Q_{4}\), for a given load condition. The four-coil link can provide an additional degree of freedom (\(k_{12}\)) from the three-coil link for the impedance matching on the source side based on (3). Since the two-, three-, and four-coil inductive links have different strengths and weaknesses in the coupling coefficient (\(k\)), PDL, and coupling variations as summarized in Table 1, the designers can select the appropriate inductive link configuration depending on the specifications of the WPT system [9]. The optimization procedures of two-, three-, and four-coil inductive links start with the design constrains imposed by the application and coil fabrication technology. The design constrains in the Rx defines the maximum outer diameter of coils in the IMD, and the coil fabrication technology indicates the minimum line width and line spacing.