## 3 Quality Factor Analysis of Air-Core Inductors
### Winding Structure Optimization of Planar Spiral Inductors  
For a planar spiral inductor, the outer diameter, \(D\), inner diameter, \(d\), coil turn, \(N\), coil-wire thickness and width determine its profile and inductance value. Inductance values of planar spiral inductors with different winding structures are uniformly expressed as follows [40]:  
\[L=q_{1}u_{0}N^{2}(D+d)\Big{[}\text{ln}\Big{(}\frac{q_{2}}{T}\Big{)}+q_{3}T+q_{4 }T^{2}\Big{]},T=\frac{D-d}{D+d}, \tag{13}\]  
where \(D\) and \(d\) are the outer diameter and inner diameter, respectively, \(T\) is the ratio of difference and sum between outer and inner diameters, and \(N\) is the coil turn. The coefficients \(q_{1}\), \(q_{2}\), \(q_{3}\) and \(q_{4}\) of different winding structures are given in Table 3.  
Considering skin effect and coil length, the \(n\)th harmonic \(AC\) resistance values of planar spiral inductors are given by  
\[R_{AC,n}=\rho\frac{I}{\varphi\theta_{n}} \tag{14}\] \[\delta_{n}=\sqrt{\frac{\rho}{n\pi f_{s}\mu_{0}}},l=p(D+d)N\,\]  
where the coefficient, \(p\), of different winding structures are as given in Table 3.  
Combining (13) and (14), the quality factor of planar spiral inductors is derived as  
\[Q_{planar}=2\varphi\sqrt{\frac{\pi\mu_{0}f_{s}\rho}{n}}\frac{q_{1}N}{p}\Big{[} \text{ln}\Big{(}\frac{q_{2}}{T}\Big{)}+q_{3}T+q_{4}T^{2}\Big{]}. \tag{15}\]  
According to (15), the quality factor depends on the cross-section perimeter, \(\varphi\), coil turn, \(N\), and \(T\). For \(\varphi\) and \(N\), the quality factor increases monotonically with them. For \(T\), the function-related \(T\) in \(Q\) is defined as  
\[F(T)=\text{ln}\Big{(}\frac{q_{2}}{T}\Big{)}+q_{3}T+q_{4}T^{2}. \tag{16}\]  
Figure 7: Structure schematics of air-core inductors: (**a**) planar spiral inductor (**b**) solenoid inductor.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Winding Structure** & **Square** & **Hexagon** & **Octagon** & **Circle** \\ \hline \(p\) & 2 & 1.732 & 1.657 & 1.571 \\ \(q_{1}\) & 0.3175 & 0.2725 & 0.2675 & 0.25 \\ \(q_{2}\) & 2.07 & 2.23 & 2.29 & 2.46 \\ \(q_{3}\) & 0.18 & 0 & 0 & 0 \\ \(q_{4}\) & 0.13 & 0.17 & 0.19 & 0.19 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Coefficients \(p\), \(q_{1}\), \(q_{2}\), \(q_{3}\) and \(q_{4}\) in different winding structures of planar spiral inductors.  
Based on (16), the relationships between \(F\) and \(T\) are plotted in Figure 8 (\(T<1\)). According to Figure 8, at the same \(D\) (area constraint), increasing \(d\) is beneficial for improving quality factor, \(Q\).  
In general, the outer diameter \(D\) is constrained by the inductance value. In order to improve the quality factor, \(d\), \(N\) and \(\varphi\) should increase as much as possible. However, the above measures are contradictory in the winding structures of equal width. Therefore, in [41], wide outside winding and the narrow inside winding are applied to improve the quality factor, as shown in Figure 9. The quality factor of the proposed winding structure is calculated using the Greenhouse algorithm. Each turn of the winding structure is approximated as a polygon. The inductance value is the sum of self inductances and mutual inductances of all turns. The AC resistance value is the sum of AC resistance values for all turns. Therefore, the quality factor can be calculated based on the inductance and AC resistance values.