## 2 Description of the Calculable Resistors  
### CRs Geometries and Model Descriptions  
To model a CR as a transmission line, wire loops are placed in the middle of the cylindrical conductor shield in a way that each wire segment is located at the same distance from the shield axis. As shown in Figure 2a, the bifilar geometry consists of a single wire loop [27]. Figure 2b illustrates the quadrifilar geometry consisting of a double loop arranged so that the current flows in opposite directions in each wire segment [2]. As displayed in Figure 2c, the octofilar geometry is like  
Figure 1: Assembly of CR pieces of: (**a**) 1 k\(\Omega\) bifilar, (**b**) 1 k\(\Omega\) quadrifilar, and (**c**) 10 k\(\Omega\) octofilar. the quadrifilar geometry but with four-wire loops in series that tend to cancel magnetic fluxes [3]. The reason why there are geometries with more wire loops is that the total length of the resistor is reduced, allowing to develop standards with higher nominal resistance values. For example, if the length of a 1 k\(\Omega\) bifilar CR is one meter (two meters of wire), the octoifilar geometry with the same two-meter wire segment will be 25 cm. The frequency performance of a resistor depends on several factors, as has been correctly explained in previous works [2; 3; 25; 28]; the presence of capacitance, inductance, and conductance between wire segments as well as among wire and shield, mostly affect the frequency dependence. As an example, Figure 2d exhibits the parasitic impedances present in a bifilar CR, where \(L\) is the self-inductance of each wire segment, \(C_{0}\) and \(G_{0}\) are the total capacitance and conductance among the wire loop and the shield, \(M\), \(C_{1}\), and \(G_{1}\) are the mutual inductance, capacitance, and conductance between wire segments respectively. For the quadrifilar and the octoifilar geometries, the mutual inductance, capacitance, and conductance among each pair of wire segment needs to be taken into account; hence, the quadrifilar CR will have \(M_{1}\) and \(M_{2}\) mutual inductance, \(C_{1}\) and \(C_{2}\) capacitances, and \(G_{1}\) and \(G_{2}\) conductances between wires, and the octoifilar CR will have \(M_{1}\), \(M_{2}\), \(M_{3}\), and \(M_{4}\) mutual inductances, \(C_{1}\), \(C_{2}\), \(C_{3}\), and \(C_{4}\) capacitances, and \(G_{1}\), \(G_{2}\), \(G_{3}\), and \(G_{4}\) conductances between wires  
According to the calculations described in [2; 3], the models that describe the change in resistance as a function of the angular frequency (\(\omega\)) due to the parasitic impedances in the CRs are:  
\[\frac{R_{ac}}{R_{dc}}= \bigg{\{}1+\frac{R}{6}(G_{0}-2G_{1})+\Big{(}\frac{\omega}{R} \Big{)}^{2}\left(L-2M_{1}\right)^{2}+\frac{\omega^{2}R^{2}}{720}\Big{[}15C_{0 }^{2}-\left(C_{0}+4C_{1}\right)^{2}\Big{]}\bigg{\}} \tag{1}\]  
for the bifilar geometry presented in Figure 2a,  
\[ \begin{split}\frac{R_{ac}}{R_{dc}}=&\bigg{\{}1+ \frac{R}{6}(G_{0}-5G_{1}-3G_{2})+\Big{(}\frac{\omega}{R}\Big{)}^{2}\left(L-8M _{1}+4M_{2}\right)^{2}\\ &+\frac{\omega^{2}R^{2}}{11520}\Big{[}240C_{0}^{2}-\left(C_{0}+ 16C_{1}\right)^{2}-15\left(C_{0}+8C_{1}+8C_{2}\right)^{2}\Big{]}\bigg{\}} \end{split}  \tag{2}\]  
Figure 2: Types of calculable resistor geometries. **(a)** Bifilar geometry; **(b)** Quadrifilar geometry; **(c)** Octtoifilar geometry; **(d)** Parasitic impedance in a bifilar CR. for the quadrifilar geometry showed in Figure 2b, and  
\[\frac{R_{ac}}{R_{dc}}= \bigg{\{}1+\frac{R}{192}(17G_{0}-128G_{1}-144G_{2}-128G_{3}-48G_{4} )+\] \[\left(\frac{\omega}{R}\right)^{2}\left(L-16M_{1}+16M_{2}-16M_{3}+8 M_{4}\right)^{2}+\frac{\omega^{2}R^{2}}{737280}\left[1920C_{0}^{2}-4\left(C_{0}+32C_ {1}+32C_{3}\right)^{2}\right. \tag{3}\] \[\left.-60\left(C_{0}+16C_{1}+32C_{2}+16C_{3}\right)^{2}-15\left(C _{0}+16C_{1}+16C_{2}+16C_{3}+16C_{4}\right)^{2}\right]\right\}\]  
for the octoifilar geometry exhibit in Figure 2c. The value of the parasitic impedances present in the CRs is calculated through their dimensions.