**3. A generalized approach to the calculation of the basic AMB and HMB overall dimensions**

In view of prospects for using of AMB and HMB, it is useful to consider the approach for the calculation of their overall dimensions in more detail.

In view of the design similarity of AMB and HMB (HMB, in which PM are used to create an additional magnetic flux), the development of a generalized approach for the AMB and HMB calculation is proposed.

To solve this problem, consider HMB design with radial or axial magnetic inserts. The fundamental difference of these designs is the arrangement of the PM for reinforcing the magnetic flux in the way of the magnetic field line. Thus, these design differences have no significant influence on the mathematical description of the HMB. Moreover, one can get AMB, equating the energy characteristics of a PM to zero that allow making a conclusion about the generalization of considered designs for AMB and HMB.

The following assumptions are used in solving the problems:


Based on the terms of the problem, the developed generalized approach should take into account both the thermal and electromagnetic processes in HMB. Therefore, the equivalent circuit method (equivalent circuits) has been selected for HMB research that is widely used in the electromagnetic and thermal processes calculations. **Figure 5** shows the equivalent circuit of the magnetic (a) and thermal circuit (b) of HMB.

The strength of the HMB determined as:

*f* = *q*(*E* + [ *v* × *B*]). (7)

This type of bearings has broad prospects for use in HS EM. For example, the Swiss company Seleroton has developed ultra-high-speed vacuumed motor CM-AMB-400 using this type of

Using the suspension based on the Lorentz forces in the electric motor in conjunction with vacuum allowed to almost completely solve the problems of the rotor friction of the air and

**3. A generalized approach to the calculation of the basic AMB and HMB** 

In view of prospects for using of AMB and HMB, it is useful to consider the approach for the

In view of the design similarity of AMB and HMB (HMB, in which PM are used to create an additional magnetic flux), the development of a generalized approach for the AMB and HMB

To solve this problem, consider HMB design with radial or axial magnetic inserts. The fundamental difference of these designs is the arrangement of the PM for reinforcing the magnetic flux in the way of the magnetic field line. Thus, these design differences have no significant influence on the mathematical description of the HMB. Moreover, one can get AMB, equating the energy characteristics of a PM to zero that allow making a conclusion about the general-

the friction in the bearing supports. Overall efficiency of the EM reaches 91–92%.

bearings (power of 250 W, the rotor speed of 400 000 rpm).

calculation of their overall dimensions in more detail.

ization of considered designs for AMB and HMB.

The following assumptions are used in solving the problems:

**overall dimensions**

**Figure 6.** The electrostatic support.

86 Bearing Technology

calculation is proposed.

$$F = \frac{pl\pi}{8} \frac{B\_b^2}{\mu\_o} \tag{8}$$

where *p*—number of poles; *l*—active length of HMB; τ <sup>=</sup> \_\_\_ *πD* <sup>2</sup>*<sup>p</sup>* —pole pitch; *B<sup>δ</sup>* —flux density in the HMB air gap.

According to the equivalent circuit from the total current law, it should be:

$$F\_w + 2Iw = 2F\_\delta + 2F\_z + F\_{\parallel} + 2F\_w + F\_{\parallel r} \tag{9}$$

where *Fm*—m.m.f of the PM; *F*<sup>δ</sup> —m.m.f of the air gap; *Fz* —m.m.f. of the stator magnetic core teeth; *Fj* —m.m.f. of the stator magnetic core back; *Fzr*—m.m.f. in the radial length of the rotor; *Fjr*—m.m.f. in the axial length of the rotor.

Taking into account that *F*<sup>δ</sup> <sup>=</sup> \_\_<sup>1</sup> μ0 *<sup>B</sup>*<sup>δ</sup> δ, then:

\*\*Taking into account that  $\mathsf{F}\_{s} = \frac{1}{\mu\_{s}} \mathsf{B}\_{s} \mathsf{S}\_{r}$  then:

$$\frac{1}{\mu\_{s}} \mathsf{B}\_{s} \mathsf{S} = \frac{\mathsf{F}\_{w} + 2lw - 2\mathsf{F}\_{z} - \mathsf{F}\_{f} - 2\mathsf{F}\_{w} - \mathsf{F}\_{r}}{2},\tag{10}$$

M.m.f. of the PM is defined as follows:

$$F\_m = H\_{cl} l\_n \tag{11}$$

Taking into account the temperature dependence of the energy characteristics of PM:

$$F\_m = H\_{cl} l\_n \tag{11}$$

Taking into account the temperature dependence of the energy characteristics of PM:

$$F\_m = H\_{cb} l\_n \left(1 - \frac{k\_{th}(\Theta\_{\rm fm} - 20)}{100}\right), \tag{12}$$

where *HcB* (Θ )—RMS values of the coercive force of the PM; ΘPM—the temperature of the PM; *kHc*—tension temperature coefficient.

It should be noted that the tension temperature coefficient can be assumed to be constant only when the temperature of the PM is 60–80°C (for intermetallic alloys *NdFeB* and *SmCo*). At temperatures outside this range, this ratio has a nonlinear dependence.

The PM temperature in the steady state operation of the HMB is determined on the basis of the thermal equivalent circuit, **Figure 7**b.

Functions approximating the actual magnetization curve of soft magnetic material from which the HMB magnetic core and shaft are made is used when taking into account the HMB magnetic core saturation:

$$H\_z = \alpha\_i \text{sh}\,\beta\_i \, B\_z \tag{13}$$

$$H\_{\rangle} = \alpha\_i \text{sh}\_i \beta\_i B\_{\rangle} \tag{14}$$

$$H\_{\rm nr} = \alpha\_2 \text{sh} \,\beta\_2 \,\mathcal{B}\_{\rm nr} \tag{15}$$

$$H\_{\flat} = \alpha\_2 \text{sh} \,\beta\_2 B\_{\flat} \tag{16}$$

where *α*<sup>1</sup> , *β*<sup>1</sup> —approximation coefficients for the soft magnetic material of the HMB magnetic core; *α*<sup>2</sup> , *β*<sup>2</sup> —approximation coefficients for the soft magnetic material of the shaft; *Bz* —flux density in the magnetic core teeth; *Bj* —flux density in the magnetic core back; *Bzr*—flux density on the shaft in the radial direction and *Bjr*—flux density on the shaft in the axial direction.

Then, using the obtained expression and real magnetization curve of the HMB magnetic core material, it is possible to create HMB characteristic taking into account the saturation (the dependence of the force of gravity from the current).

In **Figure 8**, as an example, dependence of the force from a current is made based on the saturation and for various ambient temperatures. All dependencies are built in static mode, transient thermal and electromagnetic processes when making the dependencies were not considered.

**Figure 7.** Equivalent circuit of the magnetic circuit HMB: (a) equivalent circuit of the magnetic circuit; (b) equivalent circuit of the thermal circuit. Here, *Fm*—m.m.f. of the PM; *I*—current in the AMB winding; *w*—AMB winding number turns; *Rm*—the magnetic resistance of the PM; *Rj* —the magnetic resistance of the AMB magnetic core back; *Rz* —the magnetic resistance of the AMB magnetic core teeth; *R*<sup>δ</sup> —the magnetic resistance of the HMB air gap; *Rzr*—the magnetic resistance of the rotor radial length; *Rjr*—the magnetic resistance of the rotor axial length; *R*δ*<sup>s</sup>* —the magnetic resistance of the air gap scattering; *Rms*—the magnetic resistance of the PM scattering; *Ri* —thermal resistance of the winding insulation; *Rst*—thermal resistance of the stator; *RPM*—thermal resistance of the PM insertion.

The PM temperature in the steady state operation of the HMB is determined on the basis of

Functions approximating the actual magnetization curve of soft magnetic material from which the HMB magnetic core and shaft are made is used when taking into account the HMB

—approximation coefficients for the soft magnetic material of the HMB magnetic

—approximation coefficients for the soft magnetic material of the shaft; *Bz*

sity on the shaft in the radial direction and *Bjr*—flux density on the shaft in the axial direction. Then, using the obtained expression and real magnetization curve of the HMB magnetic core material, it is possible to create HMB characteristic taking into account the saturation (the

In **Figure 8**, as an example, dependence of the force from a current is made based on the saturation and for various ambient temperatures. All dependencies are built in static mode, transient thermal and electromagnetic processes when making the dependencies

**Figure 7.** Equivalent circuit of the magnetic circuit HMB: (a) equivalent circuit of the magnetic circuit; (b) equivalent circuit of the thermal circuit. Here, *Fm*—m.m.f. of the PM; *I*—current in the AMB winding; *w*—AMB winding number

resistance of the rotor radial length; *Rjr*—the magnetic resistance of the rotor axial length; *R*δ*<sup>s</sup>*

insulation; *Rst*—thermal resistance of the stator; *RPM*—thermal resistance of the PM insertion.

of the air gap scattering; *Rms*—the magnetic resistance of the PM scattering; *Ri*

sh *β*<sup>1</sup> *Bz* (13)

sh *β*<sup>1</sup> *Bj* (14)

sh *β*<sup>2</sup> *Bzr* (15)

sh *β*<sup>2</sup> *Bjr* (16)

—flux density in the magnetic core back; *Bzr*—flux den-

—the magnetic resistance of the AMB magnetic core back; *Rz*

—the magnetic resistance of the HMB air gap; *Rzr*—the magnetic

—flux

—the

—the magnetic resistance

—thermal resistance of the winding

the thermal equivalent circuit, **Figure 7**b.

*H*<sup>z</sup> = *α*<sup>1</sup>

*Hj* = *α*<sup>1</sup>

*H*zr = *α*<sup>2</sup>

*Hjr* = *α*<sup>2</sup>

dependence of the force of gravity from the current).

density in the magnetic core teeth; *Bj*

turns; *Rm*—the magnetic resistance of the PM; *Rj*

magnetic resistance of the AMB magnetic core teeth; *R*<sup>δ</sup>

magnetic core saturation:

where *α*<sup>1</sup>

88 Bearing Technology

core; *α*<sup>2</sup>

, *β*<sup>1</sup>

were not considered.

, *β*<sup>2</sup>

**Figure 8.** Dependence of the HMB tractive force from a current value (taking into account changes in ambient temperature and magnetic core saturability). Here, 1—at a temperature of 20°C; 2—at a temperature of 60°C; 3—at a temperature of 90°C.

From these curves, it is seen that the HMB is losing its control in the magnetic core saturation area and at high temperatures. This is due to a significant nonlinearity dependence of HMB forces from the current and the magnetic flux of the PM. AMB and HMB control system is usually built on the linearization of these dependencies. Loss controllability area occurs at 0.8 A. In this area, HMB tractive force remains practically unchanged as current increases, since the magnetic core reaches saturation. At a significant saturation, HMB tractive force is slightly reduced, which causes a significant increase in stator back and teeth m.m.f. Steel 2421 was used for making dependence.
