## 7 Method Application 2: Iron Losses Computation
### Steinmetz Formulation for Iron Losses Computation  
The Steinmetz formulation has the following expression:  
\[P_{\rm fe}=\gamma_{\rm lam}\cdot\sum_{\rm el}V_{\rm el}\cdot\sum_{\
u}\left(k_{ \rm hy,\
u}\,vf_{\rm s}\,\,\hat{B}_{\rm el,\
u}^{2}+k_{\rm ec,\
u}\,\left(vf_{ \rm s}\right)^{2}\,\hat{B}_{\rm el,\
u}^{2}\right) \tag{23}\]  
with:  
\(\gamma_{\rm lam}\) \(\gamma_{\rm lam}\) \(\gamma_{\rm lam}\)  
Thus, with flux density harmonics with such far values of frequency, it is not proper to consider only one couple of \(c_{\rm hy},c_{\rm ec}\) coefficients for all the harmonics. The correct approach should be assigning to each harmonic a coherent couple of coefficient [15], interpolating the electrical steel datasheet with the closest frequencies to the considered one. The motor steel is a M400-50A, and the available specific iron losses datasheet includes the frequencies 50, 100, 200, 400, 1000 and 2500 Hz. For the fundamental, the hysteresis and eddy current coefficients are derived, interpolating the 50 and 100 Hz losses curves at 1 T. Using these coefficients, the approximation of the datasheet curve at 50 Hz is shown in Figure 14a. The adopted Steinmetz-specific losses formulation is:  
\[p_{\rm fe}=c_{\rm hy}\,B^{2}\,f+c_{\rm ec}\,B^{2}\,f^{2} \tag{27}\]  
To derive the coefficients for the high frequency harmonics, the specific losses curves at 1000 and 2500 Hz have been interpolated at 1 T. The approximations of the datasheet curves, using the high frequencies coefficients, are reported in Figure 14b. It can be noticed that the approximations in Figure 14b are good at least up to 1 T, and a greater amplitude of the flux density variation, due the slot harmonics, is not expected in this motor, except for a current much greater than the rated one. In the IM, also the stator belt harmonics (5 th, 7 th, 11 th, 13 th, etc.) produce rotor iron losses, rotating, with respect to the rotor, at a higher frequency than the stator one, function of the harmonic order and the slip [16]. However, the belt harmonics' amplitude is remarkably lower than the slot harmonics and the frequency of the most important of them (5 th and 7 th) is not much higher than the fundamental. As will be shown later, the iron losses due to the stator belt harmonics are quite low and, for the sake of simplicity, the fundamental hysteresis and eddy current coefficients are assigned to these harmonics. Figure 14: Hysteresis and eddy current coefficient derived for the fundamental and the high frequency harmonics. The approximation of the datasheet-specific losses behavior is reported. (**a**) Hysteresis and eddy current coefficient for the fundamental flux density harmonic; (**b**) hysteresis and eddy current coefficient for the high-frequency flux density harmonics. In Table 4, the stator and rotor harmonic losses are reported in the rated no-load and on-load operating points. Considering, at first, the no-load regime, it can be noticed that a serious amount of the total iron losses is due to the stator slot harmonics considering both the magneto-motive force harmonics and the slot openings. Furthermore, at no load, the stator losses due to the rotor slot harmonics, are almost negligible. Moving to the on-load operating point, the harmonics in the rotor become non-integer, due to the presence of the slip. The iron losses due to the stator winding belt harmonics are linked to the harmonics 5.82 and 11.65, applying (24) (at no-load, the losses due to these harmonics, are negligible). For the same reason, the stator slot harmonics in the rotor become of order 34.96, 69.92, etc. With the load current flowing in the rotor cage, the stator iron losses due to the rotor slot harmonics become an important amount of the total losses. Considering the current Standard definition, these kinds of losses are part of the on-load additional losses. During the on-load operation, the first harmonic losses are lower than in the on-load condition because of the voltage drop on the stator impedance, considering that the stator is supplied by a constant voltage source.