The rotor is built with laminated iron, and the iron losses are much lower compared to a solid rotor, but still should be taken into account carefully due to the poor rotor heat dissipation in the flywheel system. The rotor iron losses calculated here include two parts: the one caused by the biased field and the one caused by the control field. These two loss components are calculated separately and summed up afterwards to obtain the total losses.
1) Rotor losses caused by biased field
The biased field distribution on the rotor surface is not ideally constant. Field sags can be seen in Fig. 6-10 a) in the air gap under the stator pole gap. After Fourier transform, the field can be described by (7-19) in stator reference frame. This field is non-rotating, therefore, in the rotor reference frame, all of these field harmonics have the same rotat-ing speed relative to the rotor, which will induce rotor losses.
,B,s , ,B s
1
cos
B B p
(stator reference frame) (7-19)
,B,r , ,B r
1
cos
B B p p t
(rotor reference frame) (7-20) is the harmonic order, p is the pole counts, s and r are the angles in stator and rotor reference frames, is the phase angle of each harmonic , B,,B is the amplitude of each harmonic .
The rotor losses can be calculated numerically in JMAG. Usually a 3D transient simula-tion is required due to the unique 3D field distribusimula-tion of a homo-polar magnetic bear-ing, which is rather time consuming. Therefore, a static model is used here as an alter-native approach. Firstly, the static field calculation is performed in a 3D static model, as shown in Fig. 7-6. Then the rotor iron is segmented into small rings as shown in Fig.
7-22, with the created r--z reference frame and a sample point A at the center position of one segmented ring. The aim is to obtain the spatial field distribution versus angle at the circle which is coincident with point A and it is used to represent the field distri-bution in this segmented ring.
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A
r
z 0
Fig. 7-22 Assumed segmented rotor core in order to obtain the field distribution (versus angle ) in each segmented ring for rotor iron loss calculation based on the field
calcu-lation results in JMAG
Fig. 7-23 a) shows the field distribution in one ring of the simulation model as an ex-ample. By replacing with t in Fig. 7-23 a), this spatial field distribution is convert-ed to time domain in the rotor reference frame, which represents the field variation ver-sus time of the sample point A in the rotor in Fig. 7-22. After performing the Fourier transform in time domain, the time harmonic field can be obtained (Fig. 7-23 b)) with the field amplitude B and frequency f for each harmonic order .
a) b)
Fig. 7-23 Calculated field distribution and harmonics in one segmented ring of the rotor iron of the combined magnetic bearing: a) radial field Br under one pole pitch at the circle r = 33.603 mm, z = 9.145 mm, b) harmonics of radial field Br, base frequency
f1 = p n = 1600 Hz at speed n = 24000 min-1 with the pole number p = 4 0
0.2 0.4 0.6 0.8
0° 15° 30° 45° 60° 75° 90°
Radial field Br [T]
Angle
0 0.1 0.2 0.3 0.4 0.5 0.6
0 2 4 6 8 10 12 14 16 18 20 22 Field amplitude Br [T]
Harmonic order Segmented
rotor rings
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Finally, by using B and f, the specific losses in each segment are calculated by classical specific iron loss formula (7-21) [58]. Due to the symmetric configuration, all the points of one segmented ring have the same field spectrum after Fourier transform. Therefore, the total iron losses (in W/kg) in the rotor can be calculated by (7-22), where mseg (in kg) is the mass of each segmented ring.
2 2 2 1.5 1.5
Fe, seg h c e
1 n
p k B f k B f k B f
(7-21)Fe Fe, seg seg
seg
P
p m (7-22)In (7-21), kh, kc, ke are the iron loss coefficients considering hysteresis loss in W/(kgT2Hz), eddy current loss in W/(kgT2Hz2) and excess loss in W/(kgT1.5Hz1.5).
These coefficients can be determined by performing the curve fitting for a given loss table of certain iron sheet type. For the used iron sheet NO20 in this design, the coeffi-cients in Table 7-7 are obtained based on the loss table in Appendix B. The root mean squared error (RMSE) calculates the square root of the quadratic mean values of the differences between predicted values and observed values. It evaluates the accuracy of the predicted values calculated by (7-21) compared to the given data in the loss table.
The calculated RMSE value for the fitting is 3.985 W/kg, which is scale dependent. But compared to the absolute losses especially at high frequency (hundreds of W/kg), the RMSE is small enough and acceptable.
Table 7-7 Calculated iron loss coefficients for iron sheet NO20, obtained by data fitting of the loss table in Appendix B (RMSE: root mean squared error)
Coefficient kh
[W/(kgT2Hz)] kc
[W/(kgT2Hz2)]
ke
[W/(kgT1.5Hz1.5)]
RMSE [W/kg]
Value 0.03395 9.93910-6 4.35310-4 3.985
The calculated static field is already obtained in Fig. 7-6 for the combined magnetic bearing and in Fig. 7-16 for the radial magnetic bearing. The rotor irons, taken out from these two graphs, are presented in Fig. 7-24. The mesh size is 0.3…0.5 mm.
140
a) b)
Fig. 7-24 Static field distribution in the rotor irons, calculated in JMAG: a) in combined magnetic bearing, b) in radial magnetic bearing
For the rotor iron loss calculation, the rotor iron in combined magnetic bearing is divid-ed into 40 segmentdivid-ed rings in radial direction and 50 segmentations in axial direction.
In the radial magnetic bearing, the rotor iron is divided into 30 segmented rings in radi-al direction and 40 segmentations in axiradi-al direction. The size of each division is close to the mesh element size. After getting the losses, one correction factor kvy = 1.5 is used considering the effect of bridging between the iron sheets due to the insulation damage during the laser cutting process. The calculated losses of two magnetic bearings are shown in Table 7-8.
Table 7-8 Analytically calculated losses in the rotor iron of magnetic bearings caused by biased field, which is obtained in JMAG
Rotational speed n [min-1] Losses in rotor iron caused by biased field [W]
Combined MB Radial MB
0 0 0
6000 2.20 2.57
12000 5.82 6.78
18000 10.50 12.21
24000 16.08 18.71
2) Rotor losses caused by control field
The second part of the rotor losses is due to the spatial harmonics of the control field by feeding a sinusoidal current. (The losses caused by time harmonics are calculated in Chapter 7.4.4.3.) The control field distribution (e.g. Fig. 7-12 b) and Fig. 7-18 b)) can be described by a sum of spatial harmonics in (7-23) in stator reference frame. These
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141
harmonics rotate with the speed / , which are asynchronous with the rotor for 1 and induce rotor losses.
,c , ,c s s,1
1
cos
B B p t
(stator reference frame) (7-23)In (7-23), is the harmonic order, p is the pole counts, s is the angle in stator reference frame, is the phase angle for -th harmonic, B,,c is the amplitude for -th harmonic.
The rotor losses caused by the control field can be calculated by a 2D model without the biased field, thus neglecting the saturation effect due to the superposition of the biased field and control field. After getting the numerical results, one correction factor kvy = 1.5 is used considering the effect of the bridging between the iron sheets due to the insulation damage during the laser cutting process. For sinusoidal current feeding, the calculated rotor iron losses are shown in Fig. 7-25 for the combined magnetic bear-ing and Fig. 7-26 for the radial magnetic bearbear-ing. In the end, the losses in Table 7-8 and Fig. 7-25 and Fig. 7-26 should be summed up for the corresponding magnetic bearings to obtain the total rotor iron losses.
Fig. 7-25 Calculated rotor iron losses in the radial actuator of the combined magnetic bearing due to the sinusoidal current feeding in the control coil (current frequency f and
rotational speed n fulfill: f = n, with n in s-1), neglecting the biased field excited by the magnets, calculated by 2D model in JMAG (Displayed lines correlate to different
ampli-tudes of the applied control current.) 0
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0 6000 12000 18000 24000
Rotor iron losses [W]
Rotational speed n [min-1]
2 A 4 A 6 A 8 A
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Fig. 7-26 Calculated rotor iron losses in the radial magnetic bearing due to the sinusoi-dal current feeding in the control coil (current frequency f and rotational speed n fulfill:
f = n, with n in s-1), neglecting the biased field excited by the magnets, calculated by 2D model in JMAG (The displayed lines correlate to different amplitudes of the applied
control current.)