# Efficiency Improvement Strategy for Multiple Operating Points in Doubly Fed Magnetic Geared Motor
## 3 Iron Loss Characteristics
### Characteristics of the Magnetic Flux  
Figure 3 shows the calculation points of the radial magnetic flux density at the inner stator, outer stator, and the modulating pieces, and Figure 4 shows the time variation of the radial magnetic flux density and its time harmonic spectra at the inner stator, when the inner and outer windings operate at 180 Hz and 360 Hz (Type I), respectively. Although the time variation of the radial magnetic flux density at the inner stator has some time harmonic components, the fundamental component corresponding to the frequency of the inner winding (180 Hz) is dominant. However, as shown in Figure 5, there are two large time harmonic components of the radial magnetic flux density at the outer stator. The second harmonic which corresponds to the frequency of the outer winding (360 Hz), and the fundamental component which corresponds to the 180 Hz, are produced by the fundamental component of the magnetic flux of the inner winding. Furthermore, the amplitude of the fundamental component is almost half of the second harmonic (360 Hz). Namely, the magnetic flux of the inner winding, which has a small number of pole pairs, passes through the outer stator, and it affects the flux density variation and resultant iron loss at the outer stator [16]. Meanwhile, the time variation of the radial magnetic flux density and its time harmonic spectra at the modulating piece are presented, as shown in Figure 6. The time variation which corresponds to 13.85 Hz, and this frequency can be derived from the relative angular velocity difference between the modulating pieces and the fundamental component of the inner or outer magnetic flux, as in Equations (5) and (6), respectively:  
\[f_{by\_inner}(Hz)=[f_{inner}/p_{inner}-(f_{inner}+f_{outer})/N_{MPs}]\times p _{inner} \tag{5}\]  
\[f_{by\_outer}(Hz)=[f_{outer}/p_{outer}-(f_{inner}+f_{outer})/N_{MPs}]\times p _{outer} \tag{6}\]  
By substituting 180 HZ for \(f_{inner}\), and 360 HZ for \(f_{outer}\), into Equations (5) and (6), the frequencies of the time variation of the radial magnetic flux density caused by the inner and outer magnetic flux are obtained as in Equations (7) and (8), respectively:  
\[+13.85 Hz=[180 Hz/4-(180 Hz+360 Hz)/13]\times 4 \tag{7}\]  
\[-13.85 Hz=[360 Hz/9-(180 Hz+360 Hz)/13]\times 9 \tag{8}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Contents** & **Value** \\ \hline Diameter (mm) & 269 \\ Stack length (mm) & 156 \\ Number of pole pairs of inner winding & 4 \\ Number of pole pairs of outer winding & 9 \\ Number of modulating pieces & 13 \\ Base speed (rpm) & 2492 \\ Current density of wire (A\({}_{\text{rms}}\)/mm\({}^{2}\)) & 10 \\ Fill factor & 0.55 \\ Phase current (inner/outer), (A\({}_{\text{rms}}\)) & 424/168 \\ Number of turns per phase (inner/outer) & 16/36 \\ MMF per phase (inner/outer) (AT) & 6784/6048 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specifications of the analysis model.  
The time variation frequencies by the inner and outer magnetic flux are the same for each other. However, the positive sign, (+) of Equation (7) means that the fundamental component of the inner magnetic flux is faster than the modulating pieces, and the negative sign of Equation (8) means the modulating pieces are faster than the fundamental component of the outer magnetic flux.  
Figure 4: Variation of the radial magnetic flux densities with time at the front of tooth, center of tooth, and back yoke in the inner stator. (**a**) Waveform and (**b**) Time harmonic spectra.  
Figure 5: Variation of the radial magnetic flux densities with time at the front of tooth, center of tooth, and back yoke in the outer stator. (**a**) Waveform and (**b**) Time harmonic spectra.  
Figure 3: Calculation points of the radial flux densities in the inner stator, outer stator, and the modulating pieces.