## 3 Flywheel Electrical Model
### Formulation of the Model Parameters  
The variation of the EM internal temperature \(\theta\) significantly affects some EM parameters, and therefore the EM heating and cooling process during an operating cycle should be adequately modelled. For this purpose, a typical first order time evolution \(\theta(t)\) is supposed, according to a suitable thermal time constant \(\tau_{th}=R_{th}C_{th}\) with \(R_{th}\) and \(C_{th}\) the constant EM thermal resistance and capacitance, respectively. The parameter values are selected by examining commercial EMs with similar ratings provided with natural cooling. The \(\theta(t)\) profile will be determined by the total EM losses \(P_{l}(t)\) which are in turn affected by the temperature itself.  
The EM parameters affected by \(\theta(t)\) are the phase winding resistance \(R_{s}\) and the PM retentivity \(B_{r}\) in accordance with the PM temperature coefficient \(k_{Br}\). Such dependence impacts on both the ohmic losses \(P_{j}\) and the electromagnetic torque \(T_{em}\) according to the expressions:  
\[P_{j}=3\cdot R_{s}(\theta)\cdot I^{2}=3\ R_{s,40^{\circ}}[1+\alpha\cdot\Delta \theta_{40^{\circ}}]\cdot I^{2},\ T_{em}=k_{T}(\theta)\cdot I=k_{T,40^{\circ}} \cdot\alpha_{Br}(\theta)\cdot I \tag{6}\]  
with \(\alpha\) the temperature coefficient of the copper resistance, \(\Delta\theta_{40^{\circ}}=\theta-40\ ^{\circ}\)C, \(k_{T}\) torque constant and \(\alpha_{Br}=1-\frac{k_{Br}}{100}\Delta\theta_{40^{\circ}}\). \(R_{s,40^{\circ}}\) and \(k_{T,40^{\circ}}\) correspond to the parameter values at the reference temperature \(\theta^{*}=40\ ^{\circ}\)C. Equation (1) apply if the voltage is promptly adjusted by the current control following to the PM e.m.f. and the resistance variations and if \(k_{T}\) is not considerably affected by the armature current, as generally happens in surface mounted PM EMs. Being the dependence on current \(I\) negligible, the core losses \(P_{Fe}\) can be expressed as functions of speed \(\Omega\) and temperature \(\theta\) as:  
\[P_{Fe}=\left[P_{hys,40^{\circ}}^{*}\left(\frac{\Omega}{\Omega^{*}}\right)+P_{ cc,40^{\circ}}^{*}\left(\frac{\Omega}{\Omega^{*}}\right)^{2}\right]\cdot\alpha_{Br} ^{2}(\theta)=b_{hys}(\theta)\cdot\Omega+b_{cc}(\theta)\cdot\Omega^{2} \tag{7}\]  
with \(P_{hys,40^{\circ}}^{*}\) and \(P_{cc,40^{\circ}}^{*}\) hysteresis and eddy current losses at the reference values \(\Omega^{*}=\Omega_{max}\) and \(\theta^{*}\). Equation (7) assumes that both the contributions are proportional to the square of the flux density.  
The mechanical losses \(P_{m}\) are related to the windage and friction resistances. According to the EM-FWR arrangement, the former is only related to the EM rotor; on the contrary the latter involvesthe total suspended EM rotor-FWR mass \(M_{s}\) and depends on the bearing type and geometry. An approximate \(P_{m}\) evaluation can be obtained by the following expression:  
\[P_{m}=P_{w}+P_{fr}=a_{w}\Omega^{3}+a_{fr}\Omega\,\ a_{w}\cong 0.3D_{o}^{5} \cdot\left(1+5\frac{L_{i}}{D_{o}}\right)\cdot\left(\frac{60}{2\pi}\right)^{3} \cdot 10^{-6}\,\ a_{fr}=\mu_{b}M_{s}g\frac{d_{b}}{2}\]  
where \(D_{o}\) and \(L_{i}\) are the rotor outer diameter and length, \(d_{b}\) and \(\mu_{b}\) are the bearing diameter and friction coefficient and \(g\) is the gravitational acceleration. The semiempirical formulation of the coefficient \(a_{w}\) is valid for an EM without a fan and when speed is higher than 15,000 rpm [25]. Both \(a_{w}\) and \(a_{fr}\) are assumed to be independent of temperature. By neglecting the power consumption of the vacuum pump, the total losses are therefore given by \(P_{l}=P_{j}+P_{Fe}+P_{m}\).