## 2 Description of the Calculable Resistors
### CRs Geometries and Model Descriptions
For self-inductances and mutual inductance values, [PERSON]'s and [PERSON]'s formulas are utilized [29]. The method described by [PERSON] [30] is employed to define capacitance values. As described by [PERSON] and [PERSON] [3], the conductances can be calculated from capacitances if the permittivity and the conductivity of the medium are known. Tables 1 and 2 summarize, respectively, the dimensions of the three types of CRs developed in this research and the calculated values of their parasitic impedances. For the three types of CRs, the shields are made of aluminum tubes, and the dielectric is air. Trying to achieve the lowest resistance changes due to frequency, the models of Equations (1)-(3) were typed in a Matlab program to evaluate the CR's frequency behavior as a function of the distances between adjacent wires, shield diameter, shield thickness, and wire diameter. In this way, the geometric parameters of the RCs were selected. There are other factors such as Eddy currents that influence the frequency dependence of a CR [2], and the skin effect that changes the resistance of a cylindrical conductor due to the tendency of the current to flow through the wire surface as frequency increases. The increase in resistance owing to the skin effect is calculated from Equation (4) [3, 28]. As was established in [2], there are no significant Eddy current losses in each wire due to the proximity of the others. The increase in resistance due to the Eddy currents induced in the cylindrical shield is expressed by Equation (5) [2]. \[\frac{R_{ac}}{R_{dc}}=1+\frac{\omega^{2}}{192}\left(\frac{\mu_{0}\mu_{r}}{\pi r }\right)^{2} \tag{4}\]  
\[\frac{R_{ac}}{R_{dc}}=1+\frac{2\mu_{0}m\omega}{\pi r}\sum_{n}^{\infty}\frac{1 }{m^{2}+n^{2}}\left(\frac{b}{a}\right)^{2} \tag{5}\]  
where \(m=\frac{\omega\mu_{0}\mu_{s}as}{2\rho_{s}}\), \(\mu_{0}\) is the permeability of vacuum, \(\mu_{r}\) is the relative permeability of the wire, \(r\) is the resistance per unit of length of the wire, \(\omega\) is the angular frequency in \(rad/s\), \(b\) is the distance between wires, \(a\) is the internal shield radio, \(s\) is the shield thickness, \(\mu_{s}\) is the relative permeability of the shield, and \(\rho_{s}\) is the resistivity of the shield. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multicolumn{1}{c}{**Parameter**} & **Symbol** & **Bifilar** & **Quadrifilar** & **Octofilar** \\ \hline Shield thickness & s & 6.0 \(\pm\) 0.1 mm & 6.0 \(\pm\) 0.1 mm & 6.0 \(\pm\) 0.1 mm \\ \hline Shield diameter & a & 102.1 \(\pm\) 0.1 mm & 102.1 \(\pm\) 0.1 mm & 102.1 \(\pm\) 0.1 mm \\ \hline Wire diameter & r & 21.66 \(\pm\) 0.07 mm & 57.34 \(\pm\) 0.83 mm & 21.66 \(\pm\) 0.07 mm \\ \hline Folded length of wire & 1 & 119 \(\pm\) 0.5 mm & 443 \(\pm\) 1 mm & 291 \(\pm\) 1 mm \\ \hline Distance between adjacent wires & d & 9.4 \(\pm\) 0.5 mm & 6.7 \(\pm\) 0.5 mm & 16.1 \(\pm\) 0.5 mm \\ \hline Distance between shield axis and wires axis & b & 4.7 \(\pm\) 0.5 mm & 4.7 \(\pm\) 0.5 mm & 20.8 \(\pm\) 0.5 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions of bifilar, quadrifilar, and octoifilar CRs.