# Textile-Based Flexible Coils for Wireless Inductive Power Transmission
## 6 Electrical Characteristics of Printed Coils  
The self-resonant frequencies (SRFs) of coils are located as shown in Figure 8. The SRF of the wound copper coil was at 10.3 MHz, and the highest SRFs of both copper tape coil and the printed coil are 17.6 MHz for the un-deformed coils. A deformation of 16 m\({}^{-1}\) gives a 1% increase in the observed  
Figure 7: Measuring DC output power of Qi-standard wireless power transfer circuit with adjustable radius support for RX coil (highlighted blue) and planar support for TX coil (highlighted green). SRFs. The inductance was then measured to calculate the parasitic capacitance at 1.03 MHz and 1.76 MHz for the wound copper and flexible coils, respectively. Equivalent series resistances (ESRs), and unloaded \(Q\) factors are measured and calculated as specified in Section 5. The coil parameters are listed in Table 2 and compared with the theoretical values obtained using Equations (1)--\(L_{circular}\), (4)--\(R_{e}\), (5)--\(C_{p}\), and (6)--\(Q_{u}\). The values of practical inductance, ESR, and unloaded \(Q\) factor are calculated based on the measurements on five samples of each type of coil. The practical inductances of all types of coils are within 14% of theoretical value for flat coils. The difference in inductance between the copper tape and the printed coil is 12%, which implies the inductance of a coil is mainly affected by the winding dimensions and geometry rather than the fabrication method used. The theoretical calculation of the parasitic capacitance of printed coils from the design has a discrepancy of 17% from the experimental value obtained from the printed coil based on the measured SRF and inductance. One reason for this difference is the varying relative permittivity of the surrounding materials (i.e., the interface material and textile). The difference between the measured and theoretical \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Wound Copper Coil** & **Copper Tape Coil** & **Printed Coil** \\ \hline \(L_{coil}\) & 5.24 \(\mu\)H \(\pm\) 1\% & 4.40 \(\mu\)H \(\pm\) 2\% & 3.88 \(\mu\)H \(\pm\) 8\% \\ \(L_{circular}\) & 5 \(\mu\)H & 4.52 \(\mu\)H & 4.52 \(\mu\)H \\ \(C_{p\text{-}coil}\) & 8 pF \(\pm\) 0.5\% & 10 pF \(\pm\) 1\% & 24 pF \(\pm\) 9\% \\ \(C_{p}\) & 5 pF & 6 pF & 20 pF \\ \(R_{e\text{-}coil}\) & 6.1 \(\Omega\)\(\pm\) 2\% & 5.9 \(\Omega\)\(\pm\) 4\% & 12.0 \(\Omega\)\(\pm\) 25\% \\ \(R_{e}\) & 6 f2 & 6 \(\Omega\) & 11.3 \(\Omega\) \\ \(Q_{u\text{-}coil}\) & 10.7 \(\pm\) 2\% & 10.17 \(\pm\) 5\% & 3.05 \(\pm\) 25\% \\ \(Q_{u}\) & 10.8 & 9.37 & 4.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Electrical characteristics of coils. Figure 8: Impedance phase angle of different type of coils on frequency range from 300 kHz to 50 MHz. The self-resonant frequency can be located from this measurement as 10.3 MHz and 17.6 MHz for wound copper coil and flexible coils, respectively. ESRs of all types of coil tested are within 6%, which is within an acceptable margin. The ESR of the printed coil has the largest deviation followed by the copper tape coil on paper, and then the wound copper coil. This is due to the variation in the conductor thickness which is greatest in the printed coils when compared with the other two types of coils. Compared with the wound copper and copper tape coils, the printed coils have a reduction of up to 55% in their unloaded \(Q\) factor, which is predominantly attributable to the higher ESR of the printed flexible conductive paste compared with copper. To measure the coupling factor the printed coils are tuned in pairs as TX and RX coils for use in the WPT system and then the coupling factor \(k\) is measured following the process used by [PERSON] and [PERSON] [26]. The variation in coupling factor as a function of spacing between two coils is illustrated in Figure 9 and compared with the theoretical calculation from Equation (8) for the different coil types.