# Loss Research and Thermal Analysis of BLDC Hollow-Cup Motor Under Reactor Suppression
## 3 Improvement Design Study  
### Theoretical Foundation  
When a controller is used to power a BLDC hollow-cup motor, a current time harmonic is present in the motor winding, generating an armature magnetomotive force harmonic. The current time harmonic \(l_{k}\) is shown by Equation (1). The frequency \(f_{k}\) of the \(k\)th time current harmonic and the amplitude of the \(k\)th armature magnetomotive force harmonic generated by the \(v\)th current time harmonic are shown by Equations (2) and (3) [16], where \(f_{1}\) is the controller output current in the fundamental frequency, and \(k\) is the order of the current time harmonic [16].  
\[I_{k}=\frac{U_{k}}{2\pi f_{k}L_{a}}, \tag{1}\]  
\[f_{k}=kf_{1}, \tag{2}\]  
\[\left|F_{(k,v)}\right|=\frac{3\sqrt{2}}{\pi vp}Nk_{vv}I_{k}, \tag{3}\]  
where \(U_{k}\) is the voltage time harmonic; \(f_{k}\) is the current time harmonic frequency. The amplitude of the current time harmonic depends mainly on the armature inductance \(L_{a}\), \(\left|F_{(k,v)}\right|\) is the \(k\)th harmonic amplitude of the armature magnetomotive force generated by the current time harmonic, \(p\) is the number of motor pole pairs, \(k_{uv}\) is the winding factor for the \(v\)th harmonic of the armature's magnetic potential, and \(I_{k}\) is the RMS value of the \(k\)th current time harmonic.  
Not counting the saturation of the magnetic field, the amplitude of the armature magnetomotive force harmonic is proportional to the amplitude of its corresponding generated magnetic field, expressed as follows:  
\[\left|B_{(k,v)}\right|\propto\left|F_{(k,v)}\right|. \tag{4}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Motor Parameters** & **Numerical Value** \\ \hline Rated power & 90 W \\ Rated voltage (DCV) & 24 V \\ Rated speed & 47,000 r/min \\ Rated efficiency & 85\% \\ Frequency & 783 Hz \\ Number of poles & 2 \\ Air-gap length & 0.5 mm \\ Stator inner diameter & 20 mm \\ Stator outer diameter & 27 mm \\ Stator length & 50 mm \\ PM length & 50 mm \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the motor.  
Figure 1: A 3D Model of the BLDC hollow-cup motor.  
According to the law of magnetic induction, the induced electromotive force \(E_{(k,\
u)}\) generated in the iron core is related to the magnetic field amplitude, the alternating frequency \(f_{(k,\
u,\tau)}\), as shown in Equation (5). According to [PERSON]'s law, the eddy current \(I_{(k,\
u)}\) in the core is proportional to its induced electromotive force \(E_{(k,\
u)}\). According to [PERSON]'s law, the eddy current loss \(P_{(k,\
u)}\) generated by \(I_{(k,\
u)}\) in the core is proportional to the square of the amplitude of \(I_{(k,\
u)}\)[16]. These relationships are shown as follows:  
\[E_{(k,\
u)}\propto\left|B_{(k,\
u)}\right|\times f_{(k,\
u)\_,\tau}, \tag{5}\]  
\[I_{(k,\
u)}\propto E_{(k,\
u)}, \tag{6}\]  
\[P_{(k,\
u)}\propto I^{2}_{(k,\
u)}. \tag{7}\]  
Analysis of the above equations shows that the armature inductance \(L_{a}\) is inversely proportional to the current time harmonic \(I_{k}\). The amplitude of the current time harmonic \(I_{k}\) is proportional to the amplitude of the armature magnetomotive force harmonic \(F_{(k,\
u)}\), and the eddy current loss \(P_{(k,\
u)}\) is proportional to \(\left|F_{(k,\
u)}\right|\)[19].  
\[P_{(k,\
u)}\propto\left(\left|F_{(k,\
u)}\right|\times f_{(k,\
u)\_,\tau} \right)^{2}. \tag{8}\]