# Quasi-Wireless Capacitive Power Transfer for Wire-Free Robotic Joints +
## 2 Materials and Methods  
Conventional IPT and CPT designs require a minimum of two impedance matched Tx and Rx elements with proper alignment to allow for efficient power transfer [24]. Typically, these systems are operated in a frequency range where the currents and voltages are \begin{table}
\begin{tabular}{c c c c} \hline \hline
**IPT** & **MRPT** & **CPT** & **APT** \\ \hline \multirow{7}{*}{Pros} & Most commonly used WPT technique [18] & Employs resonance & Transfer power through & Most secure WPT \\  & & frequency matching of Tx & metallic materials without & methodology due to the \\  & & and Rx, which allows for & significant eddy current & safe transmission of sound \\  & & less reliance on alignment & losses [19; 20; 22] & waves in the required \\  & & [18; 19] & & frequency band [21] \\ \multirow{2}{*}{Pros} & High power transfer efficiency [19; 20] & Range can be extended & Less reliance on Tx and Rx & Resistivity to \\  & & using intermediate coils & alignment [18] & electromagnetic \\  & Low sensitivity to & that are tuned to the same & Implementation is often & interference due to the \\  & environmental factors & system’s resonant & more cost efficient than & transfer of energy \\  & (pollutants and weather) [21] & frequency [22; 23] & with IPT systems [20] & occurring through sound \\ \hline \multirow{7}{*}{Cons} & Potential of significant eddy current losses [19] & Complex systems, potentially consisting of additional matching & Bipolar CPT requires many & Relatively low power \\  & & & capacitive plates for & transmission levels (mW), \\ \cline{1-1}  & Potential for cross-talk due to & & networks [18] & Single-Wire CPT requires a & bio-medical devices [18] \\ \cline{1-1}  & inductance leakage [23] & High sensitivity to & large counterpoise [14] & Hardly explored in robotics \\ \cline{1-1}  & Challenging to transmit & frequency shifts due to & Normally, lower & due to low transmission \\ \cline{1-1}  & power through objects & resonance frequency & transmission efficiency and & efficiency [18] \\ \cline{1-1}  & as walls [21] & aligned Tx and Rx [18] & distance than IPT [18; 20] & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Advantages and disadvantages of various WPT configurations. uniform, allowing a lumped circuit model approach when making impedance matching calculations. However, a QWiC system operates in highly resonant mode where the distributed transmission line inductances at each section are mutually coupled to adjacent sections, forming what was defined as a mutually coupled transmission line (\(\mu\)cT-line) [25]. To reduce the impact of changes in the \(\mu\)cT-line, the approach will be to form \"taps\" along the QWR. The idea is to shunt more current into the load rather than the QWR using the tap. This form of shunt-based impedance matching attempts to create a parallel path that utilizes a capacitor to negate the effects of load on the QWR operation. The voltage distribution along an untapped QWR's \(\mu\)cT-Line, with respect to the circuit neutral, is found to be [25]  
\[V=Z_{T}I_{S}(1-e^{-\phi}) \tag{1}\]  
where \(I_{S}\) is the current flowing through the QWR, \(Z_{T}\) is the total impedance of the QWR that is exponentially distributed by \((1-e^{-\phi})\), and \(\phi\) is the position along the QWR in radians. This expression is only valid when the QWR's inductance is in resonance with the stray capacitance, which is the condition required to transfer power. The stray capacitance has the most impact at the top, or free end, of the QWR. The bottom of the QWR represents the majority of the distributed inductance. It is expected that Equation (1) will change with an applied tap. To derive the impact a tap would make on the QWR, Figure 1 is used as the distributed model of the \(\mu\)cT-line with a matching network shunted at the lower end. The total current in the QWR can be split between an upper impedance (\(Z_{LR}\)) and a lower impedance (\(Z_{LR}\)).