## 2 Materials and Methods
### Electromagnetic Analysis
As observed, the approximate values calculated on the basis of the simplified formulas differ from the values by a maximum of 3.9%. Exact values of coil resistance \(R_{c}\) and electrical power consumption \(P_{el}\) vs. the number of discs N are presented graphically in Figure 5.  
The curves presented in Figures 6 and 7 allow for a physical interpretation of the results obtained. The graphs \(z=f(N)\) for \(T_{c}=20\), 35 and 50 Nm in Figure 6a show that the number of turns \(z\) needed to a develop magnetic field in the MR fluid gaps with the average value \(B_{o}\) (it should be remembered that the excitation current in all variants is the same and equals \(I=0.6\) A) increases rapidly with the number of discs N and the number of fluid gaps 2 N and is practically independent of the values of clutching torque \(T_{c}\) (notice that the three curves in Figure 6a coincide). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Variants** & **V1** & **V2** & **V3** & **V4** & **V5** & **V6** \\ \hline \(R_{c}\) (Ohm) & 20.89 & 29.61 & 38.88 & 47.69 & 22.79 & 32.52 \\ \(R_{ca}\) (Ohm) & 20.83 & 30.35 & 39.98 & 49.16 & 22.73 & 33.65 \\ \(P_{el}\) (W) & 7.52 & 10.66 & 13.99 & 17.17 & 8.20 & 11.71 \\ \(P_{ela}\) (W) & 7.49 & 10.93 & 14.39 & 17.70 & 8.18 & 12.11 \\ \hline
**Variants** & **V7** & **V8** & **V9** & **V10** & **V11** & **V12** \\ \hline \(R_{c}\) (Ohm) & 42.22 & 51.32 & 24.55 & 35.86 & 46.49 & 56.32 \\ \(R_{ca}\) (Ohm) & 43.87 & 53.34 & 25.09 & 37.08 & 48.29 & 58.54 \\ \(P_{el}\) (W) & 15.20 & 18.47 & 8.84 & 12.91 & 16.73 & 20.27 \\ \(P_{ela}\) (W) & 15.79 & 19.20 & 9.03 & 13.35 & 17.38 & 21.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coil resistances and electrical power consumption for variants **V1–V12** (exact values: \(R_{c}\), \(P_{el}\) and approximate values determined on the basis of simplified formulas (\(R_{ca}\) and \(P_{ela}\))). Figure 5: (**a**) Values of coil resistance \(R_{c}\) and (**b**) electrical power consumption \(P_{el}\) vs. number of discs N.  
The graphs \(l_{law}=f(N)\) for \(T_{c}=\) 20, 35 and 50 Nm in Figure 6b show that as the number of discs increases, the average length of the turns of the individual coils drops, which is due to the fact that the internal diameter of the coil \(r_{ci}\) decreases (see Figure 7a, presenting \(r_{ci}=f(N)\) for \(T_{c}=\) 20, 35 and 50 Nm) and, moreover, that the coil becomes increasingly slender (see Figure 7b, presenting the thickness of the coil \(r_{ce}-r_{ci}=f(N)\) for \(T_{c}=\) 20, 35 and 50 Nm). Obviously, the rapid increase in the number of turns together with the increasing number of discs is of decisive importance and the average length of the turns (decreasing as the number of discs increases) is not able to compensate for the effects of such a rapid increase in the number of turns, the consequence of which is an increase in the resistances of the excitation windings and the electrical power consumption. Commenting finally on the results contained in Table 2 and shown graphically in Figure 5, it should be emphasized that the disadvantageous effect of a greater number of discs is an increase in the resistance values of the excitation coils, as well as an increase in clutch electrical energy consumption. The increase in copper losses is the reason why it is necessary to analyse the thermal state of clutches later in this article and to determine the influence of copper losses on the spatial temperature distributions in all the considered variants.