# Textile-Based Flexible Coils for Wireless Inductive Power Transmission
## 2 Theory of Inductive Power Transmission  
Typical inductively coupled wireless power transfer systems are based on coupled-mode theory, the resonant components being the Transmitter (TX) and Receiver (RX) coils, which are coupled in an oscillating electromagnetic field. A system diagram showing a typical arrangement is shown in Figure 1.  
The inductances of the TX and RX coils allow the transfer of power wirelessly via inductive coupling. The TX and RX coils are represented by inductors \(L_{TX}\) and \(L_{RX}\) with their equivalent series resistances (ESRs) represented by \(R_{eTX}\) and \(R_{eRX}\), and their parasatistical capacitances by \(C_{pTX}\) and \(C_{pRX}\) respectively [9]. Resonant tank capacitors \(C_{rTX}\) and \(C_{rRX}\) are connected in parallel with the TX and RX coils to tune them causing both driving and receiving circuits to resonate at same operating frequency. The load, represented by a resistance \(R_{Load}\), is powered by the rectified DC voltage from the receiver circuit. The ESRs introduced by the coupled coils waste power when the alternating currents pass through the coil, and thereby reduce the efficiency of system. These losses due to the ESRs are especially prominent in a printed coil due to the high sheet resistance of printable conductive pastes compared to bulk conductive materials. Consequently, the ESRs of printed coils are should be as low as possible. The inductance of the coils is a key parameter required to tune the system correctly, the inductance of a flat spiral coil with an air core is determined by its shape. The circular coil used in this paper is  
Figure 1: System diagram of wireless inductive power transmission system. shown in Figure 2. The engineering calculations for circular and square coils as derived in [10] are given in Equations (1) and (2) respectively:  
\[L_{circular}=\frac{\mu_{0}N^{2}d_{avg}}{2}\left(\ln\left(\frac{2.46}{\Delta} \right)+0.2\Delta^{2}\right) \tag{1}\]  
\[L_{square}=\frac{1.27\mu_{0}N^{2}d_{avg}}{2}\left(\ln\left(\frac{2.07}{\Delta} \right)+0.18\Delta+0.13\Delta^{2}\right) \tag{2}\]  
where \(\mu_{0}\) is absolute magnetic permeability, \(N\) is the number of turns the spiral coil has, \(d_{avg}=\left(d_{ext}+d_{inner}\right)/2\) is the average diameter of the flat spiral coil, and the fill ratio \(\Delta=\left(d_{ext}-d_{inner}\right)/\left(d_{ext}+d_{inner}\right)\) which are defined in [10] to simplify the effect of dimensions for a coil with external diameter \(d_{ext}\) and inner diameter dinner as shown in Figure 2. The ESRs of the coils are composed of their DC resistance, the skin effect, and the proximity effect [11]. The DC resistance of the coils can be calculated for a conductive track as:  
\[R_{\text{DC}}=\frac{R_{sheet}l_{track}}{W} \tag{3}\]  
where \(R_{sheet}\) is the sheet resistance for a given thickness, which is provided by the manufacturer; for other thicknesses the sheet resistance may be calculated as discussed in Section 4, and \(W\) is the width of the track. The ESR \(R_{e}\) has been defined in [12] as a combination of \(R_{\text{DC}}\), the skin effect, and the proximity effect, given by:  
\[R_{e}=R_{\text{DC}}\left(1+\frac{1}{1-\frac{A^{\prime}}{A}}+0.1\left(\frac{ \omega}{\omega_{crit}}\right)^{2}\right) \tag{4}\]  
where \(A^{\prime}=W\times t_{c}-(W-2\delta)(t_{c}-2\delta)\) is the reduced effective area due to skin effects for a track with thickness \(t_{c}\), and skin depth \(\delta=\sqrt{2\rho/(\omega\mu_{0}\mu_{r})}\)[13] (where \(\rho\) is resistivity and \(\mu_{r}\) is relative permeability), this reduced effective area is derived in [14] from the effective area, which is the cross section of the track \(A=W\times t_{c}\), \(\omega\) is the operating frequency, and \(\omega_{crit}=3.1(W+S)\rho/\left(\mu_{0}\mu_{r}W^{2}t_{c}\right)\) is the critical frequency [15] of a coil with track spacing \(S\) where the resistance begins to increase significantly relative to the operating frequency due to the skin and proximity effect [16].