## 2 Materials and Methods  
### Electromagnetic Analysis  
In [11; 12] the experimentally verified integrated analytical-field design method of multi-disc magnetorheological (MR) clutches was described in detail. Using this method, 12 variants of MR clutches, differing in the number of discs N and the values of the rated clutching torque \(T_{c}\) were designed and their various features and properties essential from the viewpoint of manufacturing costs and industrial applications were compared in [10]. To render this comparison fully reliable and meaningful from a technical point of view, a number of assumptions were made, the most important of which were the following:  
* The thickness of the MR fluid gap \(g\) is the same (\(g=1\) mm);
* Discs, cylinder yokes and cover yokes are made of the same magnetic steel;
* Shaft and mounting rings, separating the coil from the working region, are made of the same non-magnetic steel;
* Current density \(j_{Cu}\) is assumed the same (\(j_{Cu}=4.5\) A/mm\({}^{2}\)) [13];
* Excitation current \(I\) is assumed the same (\(I=0.6\) A);
* The magnetic flux density in MR fluid-gaps \(B_{0}\) is kept the same despite variations in the geometries of clutches (\(B_{0}=0.7\) T);
* The maximum magnetic field density \(B_{mx}\) is kept the same (\(B_{mx}=1.2\) T) despite variations in the geometries of clutches (the most saturated point lies within the cover yoke at a length approximately equal to the external radius of primary member discs). As mentioned earlier, the designed clutches were compared in terms of the following criteria: overall dimensions, geometric proportions (lengths and radii of clutches), total masses \(m\) and total volumes \(V\) and factors including clutching torque per mass ratio \(T_{c}/m\) and clutching torque per volume ratio \(T_{c}/V\). The conclusion resulting from this comparison, taking into account the costs of active materials on the one hand and the complexity of the manufacturing process on the other hand, was the statement that the optimal, recommended number of discs is N = 2. Figure 2: Exemplary two-disc MR clutch (N = 2)—general view and basic terminology. This paper, as a logical continuation of the previously presented considerations, compares exactly the same variants of MR clutches with regards to the electrical power consumption necessary to maintain the coupling state, as well as spatial temperature distribution caused by copper losses in the excitation coil with resistance \(R_{c}\). The electrical power consumption, \(P_{1}\), equal to copper losses caused by the flow of excitation current, \(I\) (Joule effect), is determined by the following formula:  
\[P_{1}=\Delta P_{Cu}=I^{2}R_{c} \tag{1}\]  
\[R_{c}=\rho\cdot\frac{l}{S_{Cu0}} \tag{2}\]  
\[S_{Cu0}=\frac{\pi\cdot d_{Cu0}^{2}}{4} \tag{3}\]  
where \(\rho\)--copper resistivity; \(l\)--wire length needed to wind the coil; \(d_{Cu0}\)--diameter of the bare conductor; \(S_{Cu0}\)--cross-section of the bare conductor. Due to the fact that the assumption of the same current density \(j_{Cu}\) in the excitation coils of various variants is made, the cross-section of the conductor \(S_{Cu0}\) is the same for all considered cases. The values of the resistances \(R_{c}\) for all point variants are, therefore, determined by the lengths of the wire \(l\) needed to wind the coils. This length is equal to the sum of the lengths of the individual turns, and these turns have different lengths depending on which coil layer they belong to (Figure 3). The geometrical cross-sections of all point variants (N = 1, 2, 3, 4 discs and \(T_{c}\) = 20, 35, 50 Nm (12 various variants)) have been presented graphically in [10] in a combined drawing, and this drawing is shown in this paper as Figure 4 with a variant numbering from **V1** to **V12**. Let us now focus on the geometrical dimensions of the clutch elements and the winding data, the knowledge of which is necessary to determine the resistances of the individual coils.