# Textile-Based Coils for Inductive Wireless Power Transmission
## 2 Materials and Methods
### Quality Evaluation of the Coils  
In the following section, we will show a method to examine the previously presented textile coils for inductive energy transmission and we will evaluate the difference between the various types. Therefore, we use the coupling factor and the quality factor. The coupling factor merely describes which fraction of a generated magnetic field actually passes through a receiving coil. To obtain a high coupling factor, it would be useful, to couple two identically constructed coils. In order to compare the transmission characteristics of the textile receiver coil, we use the transmission quality [13]. The transmission quality is a system parameter that can be determined metrologically and analytically and it enables an objective comparison of different systems. It indicates the efficiency which can be achieved under ideal conditions. It also provides information about the drop-in efficiency with variable load resistance. Transmission quality [13]  
\[Q_{k}=\sqrt{\frac{1+\frac{Im(Z_{12})^{2}}{Re(Z_{11})^{2}\cdot Re(Z_{22})}}{1- \frac{(Re(Z_{12})^{2})^{2}}{Re(Z_{11})^{2}\cdot Re(Z_{22})}}} \tag{1}\]  
Figure 6: Printed coils according to the Qi standard: Primer (**left**), printing stencil (**middle**) and printed coils (**right**). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **D180** & **D280** & **D300** \\ \hline
**Outer diameter** & 80 mm & 80 mm & 100 mm \\
**Inner diameter** & 25 mm & 23 mm & 30 mm \\
**Windings** & 11 & 7 & 7 \\
**Track width** & 1.5 mm & 3 mm & 3 mm \\
**Material** & Silver printing paste & Silver printing paste & Silver printing paste \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Structure of the wound coils according to Qi standard. We can determine the maximum possible efficiency by using the transmission quality of the system  
\[\eta_{max}=\frac{Q_{k}-1}{Q_{k}+1} \tag{2}\]  
The maximum efficiency is plotted against the transmission quality in Figure 7.  
The transmission quality was measured by using a two-port network [14]. Therefore, we developed the following equivalent circuit. The coupled coils are linear, time invariant and reciprocal systems, which can be represented in a T-equivalent circuit diagram (Figure 8). The left side of the equivalent circuit (input) is described by the voltage \(V_{1}\) and the current \(I_{1}\). Similarly, voltage \(V_{2}\) and current \(I_{2}\) describe the output. All indicated voltages and currents are alternating quantities. The impedances \(Z_{11}\), \(Z_{22}\), and \(Z_{12}\) are required. The impedance is the AC resistance consisting of an active component (ohmic resistance) and a reactive component (reactance). It establishes a relationship between voltage and current and their phase shift. \[Z=R+j\cdot X \tag{3}\]  
\(R\) represents the ohmic resistance and \(X\) represents the reactance, which is often frequency-dependent. For a coil, \(X\) with \(L\) as inductance is valid  
\[X_{L}= \omega\cdot L\text{ with }\omega=2\cdot\pi\cdot f\text{,} \tag{4}\] \[Z= R+j(2\cdot\pi\cdot f)L \tag{5}\]  
In order to determine the impedances \(Z_{11}\), \(Z_{22}\), and \(Z_{12}\), they are divided into real and imaginary parts according to (4.7). The measurement system feeds the device with a known AC voltage and measures the current in magnitude and phase. Furthermore, the  
Figure 8: T-equivalent circuit diagram. Figure 7: Quality curve: efficiency vs. transmission. system offers the possibility to convert this result directly into an inductance as well as an ohmic loss resistance according to (4.8). The first measurement corresponds to the impedance of the input, while the output is unloaded (Figure 9). This impedance is henceforth named \(Z_{1o}\). It corresponds to the impedance of the transmitting coil with a coupled receiving coil and open terminals. In this case, the impedance at the input is independent from the series impedance \(Z_{22}-Z_{12}\).