# Efficiency Improvement Strategy for Multiple Operating Points in Doubly Fed Magnetic Geared Motor
## 3 Iron Loss Characteristics
### Iron Loss According to the Frequencies of the Inner and Outer Windings  
The iron loss at each node in the 2D domain is calculated using the theory of Bertotti, and it is expressed as in (9) [17]:  
\[P_{nodle}=\Bigg{\{}k_{h}B_{m}^{2}f+\frac{1}{T}\int_{0}^{T}\!\!\!\left[\sigma\frac {d^{2}}{12}\!\left(\frac{dB}{dt}(t)\right)^{2}+k_{c}\!\left(\frac{dB}{dt}(t) \right)^{3/2}\right]\!dt\Bigg{\}}k_{f} \tag{9}\]  
where \(k_{h}\) and \(k_{c}\) are the coefficient of losses by hysteresis and excess, \(\sigma\) is the electric conductivity and \(d\) is the thickness of the lamination, \(f\) and \(B_{m}\) are the electric frequency and the peak value of the magnetic flux density, \(k_{f}\) and \(T\) are the staking factor and the time period of the magnetic flux density variation, respectively. The iron loss at the individual part in the inner stator, outer stator, and modulating pieces is given by integrating the iron loss at each node over each region. However, it should be noted that the minor loop is not considered in the hysteresis loss calculation of Equation (9). Although the calculation methods that consider the minor loop or the time harmonic of the magnetic field are handled in the application which needs the exact calculation of the iron loss, it is too time consuming when the iron losses in all operating points are calculated [18,19]. Thus, the fundamental component of the time variation of the magnetic field is only considered in the hysteresis loss calculation by Equation (9), and the different frequencies derived in Figures 4-6 are used in the calculations of each part, respectively. Table 2 shows the iron loss results at the inner stator, outer stator, and the modulating pieces according to the frequencies of the inner and outer windings, when the DFMGM operates at the base speed II with the full load condition. In Table 2, whereas the iron loss at the inner stator is almost proportional to the frequency of the inner winding, the iron loss at the outer stator is not proportional to the frequency of the outer winding. In other words, the outer iron loss (602.9 W) in the Type III with the 180 Hz at the outer winding has a similar value to that (619.0 W) in the Type II with the 270 Hz at the outer winding. This is because the iron loss at the outer stator by the fundamental component of the inner magnetic flux is largely increased as the frequency of the inner winding is increasing. Furthermore, the inner iron loss (983.4 W) in Type III with 360 Hz at the inner winding are larger than the outer iron loss (701.3 W) in Type I with 360 Hz at the outer winding. Namely, under the same operating frequency, although the outer stator suffers the flux density variation by the fundamental component of inner magnetic flux, the inner stator has a greater iron loss than that of the outer stator. Hence, in the total iron loss of the ferromagnetic materials, the Type III, which has relatively high frequency at the inner winding, has the greatest iron loss. To clearly understand these tendencies, the iron loss maps over the constant torque region of Types I, II, and III are compared. It should be noted that, as mentioned above, the amplitude of the inner winding current is adjusted with respect to the torque regulation and the frequency control  
Figure 6: Variation of the radial magnetic flux densities with time at the center of modulating piece. (**a**) Waveform and (**b**) Time harmonic spectra. sequence corresponds with that in Figure 2. As shown in Figure 7c, in Type III with the high frequency at the inner winding, the iron loss is increasing as it goes to the high-torque region. This tendency is natural because the inner winding current, which acts as the armature part, is large in the high-torque region. Furthermore, as shown in Figure 7d, the iron loss map for the difference between the Type III and Type I implies that the iron losses of the Type III in the high-speed and high-torque region are greater than those of the Type I and it is reversed in the high-speed and low-torque region.