## 2 Materials and Methods
### Electromagnetic Analysis
Letter markings of the clutch geometric dimensions and winding data (the number of coil turns \(z\), the number of turns per one layer \(n\) and the number of layers \(m\)), necessary for determining the resistance of the coils, are shown in Figure 3 [10] on the example of an MR clutch with the number of discs: N = 2. The design dimensions common for all the considered variants include the following: \(\Delta_{3}\), \(d_{Cu}\), \(d_{CuO}\), \(S_{Cu}\) and \(S_{Cu0}\) (Figure 3). The design dimensions, which are different for individual variants and depend on the rated values of the clutching torque, \(T_{c}\), and the number of discs, N, are summarized in Table 1 (these values were determined on the basis of the integrated analytical-field design method described in [10,12]). Figure 3: The cross-section of clutch coil: (**a**) dimensions of coil and (**b**) dimensions of conductors. The formulas for the lengths of the wires needed to wind the coils differ depending on whether the last layer is full; i.e., whether the condition  
\[z=m\cdot n \tag{4}\]  
is fulfilled (Figure 3a) or whether the last layer is incomplete (Figure 3b), and there is a mathematical inequality. \[n\cdot m>z. \tag{5}\]  
If the condition (4) is met, we have the following:  
\[l=\sum_{i=1}^{m}\left[2\pi\cdot\left(r_{ci}+\Delta_{3}+(i-1)\cdot d_{cu}+\frac {d_{cu}}{2}\right)\right]\cdot n, \tag{6}\]  
where \(r_{ci}\) denotes the internal radius of the coil, \(\Delta_{3}\) denotes the thickness of the carcass and \(d_{Cu}\) denotes the diameter of the insulated conductor. If condition (5) is fulfilled, then another formula should be used:  
\[l=2\pi\cdot\sum_{i=1}^{m-1}\cdot\left(r_{ci}+\Delta_{3}+(i-1)\cdot d_{cu}+\frac {d_{cu}}{2}\right)\cdot n+2\pi\cdot\left(r_{ci}+\Delta_{3}+(m-1)\cdot d_{cu}+ \frac{d_{cu}}{2}\right)\cdot z_{m} \tag{7}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Variants** & **V1** & **V2** & **V3** & **V4** & **V5** & **V6** \\ \hline \(r_{ci}\) (mm) & 51.9 & 42.1 & 37.2 & 34.2 & 61.5 & 49.7 \\ \(r_{ce}\) (mm) & 83.9 & 62.3 & 56.0 & 51.9 & 85.9 & 65.9 \\ \(L_{c}\) (mm) & 4 & 10 & 16 & 22 & 5 & 13 \\ \(z\) (\(\cdot\)) & 366 & 694 & 1026 & 1365 & 368 & 695 \\ \(m\) (\(\cdot\)) & 62 & 39 & 36 & 34 & 47 & 31 \\ \(n\) (\(\cdot\)) & 6 & 18 & 29 & 41 & 8 & 23 \\ \hline
**Variants** & **V7** & **V8** & **V9** & **V10** & **V11** & **V12** \\ \hline \(r_{ci}\) (mm) & 43.9 & 40.2 & 68.7 & 55.4 & 48.9 & 44.7 \\ \(r_{ce}\) (mm) & 58.1 & 53.4 & 93.1 & 71.6 & 63.1 & 57.9 \\ \(L_{c}\) (mm) & 21 & 29 & 5 & 13 & 21 & 29 \\ \(z\) (\(\cdot\)) & 1028 & 1363 & 370 & 697 & 1030 & 1364 \\ \(m\) (\(\cdot\)) & 27 & 25 & 47 & 31 & 27 & 25 \\ \(n\) (\(\cdot\)) & 39 & 55 & 8 & 23 & 39 & 55 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of design geometrical dimensions and winding data different for 12 individual variants. Figure 4: Graphical overview of clutch cross-sections for all 12 designed variants: \(T_{c}=20,35,50\) Nm, N = 1, 2, 3, 4.  
where the number of turns in the incomplete m-th layer is \(z_{m}=z-(m-1)n\). Formulas (1)-(3), (6) and (7) allow the resistance of the excitation windings coils to be determined in an accurate manner as the sum of the resistances of the turns of the individual layers. When designing transformers and electrical machines, we very often use simplified formulas, based on the concept of the average turn length \(l_{tav}\), calculated on the basis of the overall dimensions of the coil, i.e., inner radius of the coil \(r_{ci}\) and outer radius of the coil \(r_{ce}\). \[l_{tav}=2\pi\cdot\frac{(r_{ci}+r_{ce})}{2}. \tag{8}\]  
Using the above relation, the formula for the coil resistance excitation winding resistance can be greatly simplified. In Table 2, the coil resistances \(R_{c}\) and the electrical power consumption \(P_{1}\) determined on the basis of exact Formulas (6) and (7) as well as their approximate values determined on the basis of simplified Formula (8) are listed.