2.3 Modification of Rotor Lamination
As electrical performance, back EMF and radial force have to be estimated from the
beginning of multi-physical simulation. Fig. 6 illustrates one third of cross-sectional view of the rotor, and the position of a
slit is defined for the purpose of changing the path of magnetic flux. There are three
different sets of slits, and the objective of this section is to determine which combination
of slits is more effective to noise. Before analyzing noise, key electromagnetic characteristics
such as back EMF, torque, torque ripple, and radial force are analyzed for eight models
in terms of slit combination.
Fig. 6. Different positions of slits
Two major harmonics in the back EMF of the base model are the fifth and seventh components
(15), and Fig. 7 and Table 4 give their results in the eight models with respect to slit position. Since the location
of slit #3 makes no change in components of back EMF, the four models of Base, Slit
13, Slit 23, and Slit 123 are only selected for further analysis in this paper.
Table 4. Major harmonics in back EMF with respect to slit position
|
Harmonic order
|
Base
Slit 3
|
Slit 1
Slit 13
|
Slit 2
Slit 23
|
Slit 12
Slit 123
|
|
1$^{\text{st}}$ [V]
|
128.9
|
128.3
|
128.9
|
128.5
|
|
5$^{\text{th}}$ [V]
|
10.87
|
6.53
|
14.03
|
11.55
|
|
7$^{\text{th}}$ [V]
|
3.02
|
0.72
|
5.66
|
4.14
|
Fig. 7. 5th and 7th harmonics of back EMF with respect to slit position
Fig. 8. Distribution of magnetic flux line in motor
Total force generated in air gap is decomposed into two vectors, Fr and Ft, in radial
and tangential direction as given in equations (1) and (2). The magnitude of the two
force vectors is determined by two magnetic flux densities, Br and Bt, as following.
Fig. 8 shows magnetic flux line in Base and Slit 123 models, and compared to Base, the addition
of all the slits affects the path of magnetic flux. As a result, radial force is changed
in air gap, and the objective of the following sections is to verify whether the slit
location leads to the improvement of noise.
Table 5 shows electromagnetic characteristics with respect to slit position under the same
condition of current. By adding slits to the rotor, average torque and torque ripple
deteriorate at the same time. However, peak to peak of radial force density becomes
better as shown in Table 5 and Fig. 9.
Table 5. Electromagnetic performance with respect to slit position
|
Harmonic order
|
Base
|
Slit 13
|
Slit 23
|
Slit 123
|
|
Torque
[Nm]
|
2.1259
|
2.0958
|
2.1147
|
2.0945
|
|
Torque ripple [%]
|
19.26
|
27.81
|
29.40
|
27.78
|
|
Radial force density pk2pk [$N /m^{2}$]
|
15,908
|
11,742
|
12,863
|
11,072
|
Fig. 9. Waveforms of radial force density with respect to slit position
The amplitude of vibration displacements of mode $m$ is derived as
where $M$, $\omega_{m}$, $w_{r}$, and $\zeta_{m}$ are the mass (kg) of a stator, the
angular natural frequency of mode $m$, the angular frequency of force component of
order $r$, and the modal damping ratio, respectively.
The amplitude of force is determined by
where $L_{stk}$ and $P_{mr}$ are stator stack length and the magnitude of magnetic
pressure of order $r$, respectively.
Fig. 10. Comparison of displacement between FEA and calculation results in Slit 123
When the frequency of excitation force is close to the natural frequency of the stator,
vibration reaches its resonant point and noise becomes worse significantly. In other
words, if the frequency of radial force components is close to the natural frequencies
of the stator, the components have to be considered as an important candidate to vibration
response.
Fig. 10 shows the comparison of vibration displacement between FEA and calculation results
in the model of Slit 123. The x-axis of Fig. 10 is 6X components of mechanical frequency, and they represent frequencies causing
larger vibration displacement than others. It is noted that the vibration displacement
of 360Hz is much larger than that of other frequencies. The validity of the calculation
process has been verified by the comparison of FEA within acceptable range, and the
same numerical calculation is performed in the four models of Base, Slit 13, Slit
23, and Slit 123 as shown in Fig. 11.
The left side of Fig. 11 illustrates the vibration displacement of 360Hz and 720Hz. Since the displacement
of 360Hz is much bigger than that of 720Hz, space harmonic components at 360Hz are
given in the right side of Fig. 11. It is noted that the third harmonic is the most dominant component in the four models.
In the same case of peak to peak radial force given in Table 5, the third harmonic component is smallest in Slit 123.
Fig. 11. Space harmonic components in displacement with respect to slit position at 360Hz
SPL is defined as following.
Fig. 12. Waveforms of SPL with respect to slit position
where $P_{ref}$ and $P$ are minimum audibility pressure $20\bullet 10^{-6}[Pa]$ and
sound pressure, respectively, determined by displacement $A_{m}$ and frequency $f$
as following.
where $\rho$ and $c$ are the density of air in kilograms per cubic meter and the speed
of sound in meters per second, respectively.
Since the pressure of noise is proportional to displacement, the vibration displacement
and the noise have the same tendency. In Fig. 12, noise is estimated through FEA with ideal sinusoidal current in the four models.
Fig. 12 gives SPL of the four models and the result has been summarized in Table 6. Compared to Base model, the maximum and average noise of Slit 123 are reduced by
6.89dB and 5.84dB, respectively. The order of SPL in the four cases in Table 6 is exactly same as that of radial force as given in Table 5. The location of slits is the minor design modification of the rotor, but it changes
the characteristics of noise significantly. As a result, it is noted that cautious
trade-off between torque and noise performances has to be considered during the modification
of motor lamination.
Table 6. Comparison of SPL with respect to slit position
|
SPL [dB]
|
Base
|
Slit 13
|
Slit 23
|
Slit 123
|
|
Max
|
43.77
|
40.39
|
41.29
|
36.88
|
|
Average
|
15.67
|
13.77
|
13.99
|
9.83
|