*2.1.2. Electromagnet design*

In order to design the electromagnet using the Probable Flux Paths Method, first, the attractive force derived from the magnetic circuit caused by the electromagnet needs to withstand the weight of the mass. Here, the following steps in design are taken so that sufficient attractive force from the electromagnet is produced for control.


Since the attractive force calculated through the Probable Flux Paths Method has a large error with the actual experimentation values, in order to manufacture magnetic bearings based on this design, it is desirable to design with a safety factor of greater than 3.

**Figure 5.** Electromagnet core

196 Performance Evaluation of Bearings

excluding the levitating object are designed.

*2.1.1. Probable flux paths method* 

software such as Maxwell is desirable.

or series.

*2.1.2. Electromagnet design* 

Paths Method to analyze the magnetic circuit.

smooth circular arc(quadratic curve).

a. The mass of the levitating object is determined.

of windings during normal conditions.

a. The relationship between magnetic flux and current is linear.

b. The average magnetic flux passes through the centroid of the cross section.

sufficient attractive force from the electromagnet is produced for control.

b. The material of the core and levitating object is determined.

Since a magnetic bearing system like Fig. 3 has a symmetric form vertically and horizontally, the levitating object can be simply assumed as a point mass in the perspective that the object is levitated. Therefore, in this section, the elements of the magnetic bearing system

To support the levitating object through the electromagnet's attractive force, the attraction relationship between the current in the electromagnet coil and the levitating object needs to be defined clearly. The Probable Flux Paths Method assumes that the magnetic permeability of the magnetic substance that forms the magnetic path is linear to calculate the permeance of the magnetic substance that the magnetic path passes through, followed by the calculations of the magnetomotive force, magnetic flux, magnetic flux density, and attractive force. As the permeability of materials disregarding permanent magnets are generally nonlinear, the error between the Probable Flux Paths Method calculation results and that of actual experimentation measurements is large and calculations by applying the Probable Flux Paths Method for magnetic substances with complicated magnetic paths is known to be difficult. However, since the vertically and horizontally symmetric magnetic bearing system magnetic path is of a simple form, the electromagnet is designed by applying the Probable Flux Paths Method early in the design process. For more precise designing, the use of FEM

Generally, the following assumptions have to be satisfied when using the Probable Flux

c. When the cross section that the magnetic flux passes through changes, the parts are calculated by dividing them into different parts and setting as combinations of parallel

d. When the cross section of a part changes rapidly, the magnetic flux passes through in a

In order to design the electromagnet using the Probable Flux Paths Method, first, the attractive force derived from the magnetic circuit caused by the electromagnet needs to withstand the weight of the mass. Here, the following steps in design are taken so that

c. The attractive force of the electromagnet is calculated with values determined by assumptions regarding the current, magnetic circuit, and length of the coil and number

**Figure 6.** Electromagnet magnetic circuit formation

$$\mathrm{Im}\frac{d\mathbf{v}}{dt} = \mathrm{mg} - \mathrm{F\_m} \tag{1}$$

$$\mathbf{F\_m = \frac{B^2}{2\mu\_0} A\_m} \tag{2}$$

$$\spadesuit = \frac{\text{F}\_{\text{mf}}}{\text{R}\_{\text{m}}} = \frac{\text{NI}\_{\text{m}}}{\frac{\text{lm}}{\mu\_{\text{0}}\mu\_{\text{s}}\text{S}}} = \frac{\text{NI}\_{\text{m}}}{\frac{\text{lm}}{\mu\_{\text{0}}\mu\_{\text{s}}\text{S}} + 2\frac{\text{x}\_{0}}{\mu\_{\text{0}}\text{S}}} = \frac{\mu\_{0}\text{SNI}\_{\text{m}}}{\frac{\text{lm}}{\mu\_{\text{0}}} + 2\text{x}\_{0}}\tag{3}$$

$$\mathbf{B} = \frac{\oplus}{\mathbf{S}} = \frac{\mu\_0 \text{Nl}\_\text{m}}{\frac{\text{l}\_\text{m}}{\mu\_\text{s}} + \text{Zx}\_\text{0}} \tag{4}$$

$$F\_{\rm m} = \frac{\mu\_0 \mathbf{N}^2 \mathbf{l}\_{\rm m}^2 \mathbf{S}}{\left(\frac{\mathbf{l}\_{\rm m}}{\mu\_{\rm s}} + \mathbf{z} \mathbf{x}\_0\right)^2} \tag{5}$$

$$\mathbf{F\_m} = \mathbf{k} \left( \frac{\mathbf{l\_{ss}} + \mathbf{l}}{\mathbf{x\_0} + \mathbf{w} + \mathbf{x}} \right)^2 \tag{6}$$

$$\mathbf{m}\mathbf{g} - \mathbf{k} \left(\frac{\mathbf{l\_{ss}}}{\mathbf{x\_0} + \mathbf{w}}\right)^2 = \mathbf{0} \tag{7}$$

$$\mathbf{X}\_0 = \frac{\mathbf{l\_{ss}}}{2\mu\_\mathbf{s}} \text{ and } \mathbf{k} = \frac{\mathbf{N^2\mu\_\mathbf{o}S}}{4}.$$

$$\left|\mathbf{k}\left(\frac{\mathbf{l}\_{\rm{sz}}+\mathbf{i}}{\mathbf{X}\_{\rm{0}}+\mathbf{W}+\mathbf{x}}\right)^{2}-\mathbf{k}\left(\frac{\mathbf{l}\_{\rm{z}\rm{x}}+\mathbf{i}}{\mathbf{X}\_{\rm{0}}+\mathbf{W}+\mathbf{x}}\right)^{2}\right|\_{\mathbf{i}=\mathbf{0},\mathbf{x}=\mathbf{0}}+\frac{\partial}{\partial\mathbf{l}}\mathbf{k}\left(\frac{\mathbf{l}\_{\rm{z}\rm{x}}+\mathbf{i}}{\mathbf{X}\_{\rm{0}}+\mathbf{W}+\mathbf{x}}\right)^{2}\Big|\_{\mathbf{i}=\mathbf{0},\mathbf{x}=\mathbf{0}}\left(\mathbf{i}-\mathbf{0}\right)+\frac{\partial}{\partial\mathbf{x}}\mathbf{k}\left(\frac{\mathbf{l}\_{\rm{z}\rm{x}}+\mathbf{i}}{\mathbf{X}\_{\rm{0}}+\mathbf{W}+\mathbf{x}}\right)^{2}\Big|\_{\mathbf{i}=\mathbf{0},\mathbf{x}=\mathbf{0}}\left(\mathbf{x}-\mathbf{0}\right)\Big|\_{\mathbf{i}=\mathbf{0},\mathbf{x}=\mathbf{0}}\right\rangle$$

$$\mathbf{k}\left(\frac{\mathbf{l}\_{\rm{zz}}+\mathbf{i}}{\mathbf{X}\_{\rm{0}}+\mathbf{W}+\mathbf{x}}\right)^{2}=\mathbf{k}\left(\frac{\mathbf{l}\_{\rm{zz}}}{\mathbf{X}\_{\rm{0}}+\mathbf{W}}\right)^{2}+\frac{2\mathbf{k}\mathbf{l}\_{\rm{z}}}{(\mathbf{X}\_{\rm{0}}+\mathbf{W})^{2}}\mathbf{i}-\frac{2\mathbf{k}\mathbf{l}\_{\rm{z}\rm{z}}^{2}}{(\mathbf{X}\_{\rm{0}}+\mathbf{W})^{3}}\mathbf{x}\tag{8}$$

$$\mathbf{m}\ddot{\mathbf{x}} = \mathbf{m}\mathbf{g} - \left\{ \mathbf{k} \left( \frac{\mathbf{l}\_{\rm s\overline{s}}}{\mathbf{X}\_{0} + \mathbf{W}} \right)^{2} + \frac{2\mathbf{k}\mathbf{l}\_{\rm s\overline{s}}}{(\mathbf{X}\_{0} + \mathbf{W})^{2}} \mathbf{i} - \frac{2\mathbf{k}\mathbf{l}\_{\rm s\overline{s}}^{2}}{(\mathbf{X}\_{0} + \mathbf{W})^{3}} \mathbf{x} \right\} \tag{9}$$

$$\mathbf{im\ddot{x}} = \frac{2\mathbf{k}\mathbf{l}\_{\mathbf{s}\mathbf{s}}^2}{(\mathbf{x}\_0 + \mathbf{W})^3} \mathbf{x} - \frac{2\mathbf{k}\mathbf{l}\_{\mathbf{s}\mathbf{s}}}{(\mathbf{x}\_0 + \mathbf{W})^2} \mathbf{i} \tag{10}$$

$$\frac{d}{dt}\left(\mathcal{L}(\mathbf{I}\_{\rm ss} + \mathbf{i})\right) + \mathcal{R}(\mathbf{I}\_{\rm ss} + \mathbf{i}) = \mathcal{E}\_{\rm ss} + \mathbf{e} \tag{11}$$

$$\text{RI}\_{\text{ss}} = \text{E}\_{\text{ss}} \tag{12}$$

$$L = \frac{\mathbf{N} \cdot \Phi}{\mathbf{I}\_{\text{m}}} \tag{13}$$

$$\mathcal{L} = \frac{\mathcal{Q}}{\mathcal{X}\_0 + \mathcal{W} + \mathbf{x}} \tag{14}$$

$$\mathcal{L} = \frac{\mathcal{Q}}{\mathcal{X}\_0 + \mathcal{W} + \mathcal{x}} + \mathcal{L}\_0 \tag{15}$$

$$\frac{d}{dt}\left(\mathcal{L}\left(\mathbf{I}\_{\rm ss} + \mathbf{i}\right)\right) = \mathcal{L}\frac{d}{dt}\left(\mathbf{I}\_{\rm ss} + \mathbf{i}\right) + \left(\mathbf{I}\_{\rm ss} + \mathbf{i}\right)\frac{d\mathcal{L}}{dt}$$

$$\frac{d}{dt}\left(\mathcal{L}\left(\mathbf{I}\_{\rm ss} + \mathbf{i}\right)\right) = \mathcal{L}\frac{d\mathcal{l}}{dt} + \left(\mathbf{I}\_{\rm ss} + \mathbf{i}\right)\frac{\partial\mathcal{L}}{\partial\mathbf{x}}\frac{d\mathbf{x}}{dt}$$

$$\frac{d}{dt}\left(\mathcal{L}\left(\mathbf{I}\_{\rm ss} + \mathbf{i}\right)\right) = \mathcal{L}\frac{d\mathcal{l}}{dt} + \left(\mathbf{I}\_{\rm ss} + \mathbf{i}\right)\frac{-\mathcal{Q}}{(\mathbf{X}\_{\rm 0} + \mathbf{W} + \mathbf{x})^{2}}\frac{d\mathbf{x}}{dt}\tag{16}$$

$$\mathrm{L\frac{dl}{dt} - \frac{Q}{(\mathbf{X}\_0 + \mathbf{W} + \mathbf{x})^2}\dot{\mathbf{x}}(\mathbf{I}\_{\mathrm{ss}} + \mathbf{i}) + \mathrm{R}(\mathbf{I}\_{\mathrm{ss}} + \mathbf{i}) = \mathbf{E}\_{\mathrm{ss}} + \mathbf{e} \tag{17}$$

$$\mathrm{L\frac{dl}{dt} - \frac{Q}{(\chi\_0 + \mathcal{W} + \mathbf{x})^2}\dot{\mathbf{x}}(\mathbf{I}\_{\mathbf{s}\mathbf{s}} + \mathbf{i}) + \mathrm{Ri} = \mathbf{e}}\tag{18}$$

$$\mathrm{Li}\frac{\mathrm{dl}}{\mathrm{dt}} - \frac{\mathrm{Q}}{(\mathrm{X}\_{0} + \mathrm{W})^{2}} \dot{\mathrm{x}} \mathrm{I}\_{\mathrm{SS}} + \mathrm{Ri} = \mathrm{e} \tag{19}$$

$$-\frac{R\_{\rm l}}{R\_{\rm F}}\mathcal{R}\_{\rm S}\mathcal{I}\_{\rm o} - \frac{R\_{\rm l}}{Z\_{\rm I}}(\mathcal{Z}\_{\rm L} + \mathcal{R}\_{\rm S})\mathcal{I}\_{\rm o} = \mathcal{V}\_{\rm IN}$$

$$-\left(\frac{\text{R}\_{\text{I}}\text{R}\_{\text{S}}}{\text{R}\_{\text{F}}} + \frac{\text{R}\_{\text{l}}(\text{Z}\_{\text{L}} + \text{R}\_{\text{S}})}{\text{Z}\_{\text{l}}}\right)\mathbf{I}\_{\text{o}} = \mathbf{V}\_{\text{IN}}$$

$$-\left(\frac{\text{R}\_{\text{I}}\text{R}\_{\text{S}}\text{Z}\_{\text{I}} + \text{R}\_{\text{F}}\text{R}\_{\text{l}}(\text{Z}\_{\text{L}} + \text{R}\_{\text{S}})}{\text{R}\_{\text{F}}\text{Z}\_{\text{l}}}\right)\mathbf{I}\_{\text{o}} = \mathbf{V}\_{\text{IN}}$$

$$\frac{\mathbf{I}\_{\text{o}}}{\mathbf{V}\_{\text{IN}}} = -\frac{\mathbf{R}\_{\text{F}}\mathbf{Z}\_{\text{l}}}{\mathbf{R}\_{\text{I}}\mathbf{R}\_{\text{S}}\mathbf{Z}\_{\text{I}} + \mathbf{R}\_{\text{F}}\mathbf{R}\_{\text{I}}(\text{Z}\_{\text{L}} + \text{R}\_{\text{S}})}\tag{20}$$

$$\mathbf{Z}\_{\rm I} = \frac{\mathbf{R}\_{\rm d} \mathbf{C}\_{\rm t} \mathbf{s} + \mathbf{1}}{\mathbf{c}\_{\rm t} \mathbf{s}} \tag{21}$$

$$\mathbf{Z}\_{\rm L} = \mathbf{L}\mathbf{s} + \mathbf{R} \tag{22}$$

$$\frac{\mathbf{I}\_{\rm o}}{\mathbf{V}\_{\rm IN}} = -\frac{\frac{\mathbf{R}\_{\rm F}\mathbf{d}\_{\rm C}\mathbf{t}\_{\rm f} + \mathbf{1}}{\mathbf{c}\_{\rm f}\mathbf{s}}}{\frac{\mathbf{R}\_{\rm I}\mathbf{R}\_{\rm S}\frac{\mathbf{R}\_{\rm C}\mathbf{t}\_{\rm f} + \mathbf{1}}{\mathbf{c}\_{\rm f}\mathbf{s}} + \mathbf{R}\_{\rm F}\mathbf{R}\_{\rm I}(\mathbf{L}\mathbf{s} + \mathbf{R} + \mathbf{R}\_{\rm S})}}} $$
 
$$\frac{\mathbf{I}\_{\rm o}}{\mathbf{V}\_{\rm IN}} = -\frac{\mathbf{R}\_{\rm F}(\mathbf{R}\_{\rm d}\mathbf{C}\_{\rm f}\mathbf{s} + \mathbf{1})}{\mathbf{R}\_{\rm I}\mathbf{R}\_{\rm S}(\mathbf{R}\_{\rm d}\mathbf{C}\_{\rm f}\mathbf{s} + \mathbf{1}) + \mathbf{R}\_{\rm F}\mathbf{R}\_{\rm I}(\mathbf{L}\mathbf{s} + \mathbf{R} + \mathbf{R}\_{\rm S})}} $$
 \\ \frac{\mathbf{I}\_{\rm o}}{\mathbf{V}\_{\rm IN}} = -\frac{\mathbf{R}\_{\rm F}\mathbf{R}\_{\rm d}\mathbf{C}\_{\rm C}\mathbf{s} + \mathbf{R}\_{\rm F}}{\mathbf{R}\_{\rm F}\mathbf{R}\_{\rm I}\mathbf{L}\mathbf{C}\_{\rm s}\mathbf{s}^{2} + \left(\mathbf{R}\_{\rm I}\mathbf{R}\_{\rm S}\mathbf{R}\_{\rm d} + \mathbf{R}\_{\rm I}\mathbf{R}\_{\rm F}(\mathbf{R} + \mathbf{R}\_{\rm S})\right)\mathbf{C}\_{\rm f}\mathbf{s} + \mathbf{R}\_{\rm I}\mathbf{R}\_{\rm S}}} \tag{23}

$$\mathbf{G\_{MB}} = \frac{\mathbf{b\_{mb1}s} + \mathbf{b\_{mb0}}}{\mathbf{a\_{mb2}s^4}s^4 + \mathbf{a\_{mb3}s^3}s^3 + \mathbf{a\_{mb2}s^2} + \mathbf{a\_{mb1}s} + \mathbf{a\_{mb0}}}$$

process of generating the next generation population after going through the aforementioned series of processes is described as one generation and the method to finding an optimized solution to an objective function through operations in a specific generation is defined as an algorithm.

Control of Magnetic Bearing System 205

load(here, coil). When defining the form of output desired by the designer using time response characteristics, the amplifier peripheral device values can be determined in the

First, the value of the part to solve is defined. Since the amplification ratio A of the current amplifier, current limiting resistance R�, load inductance L, and resistance R are unknown, the variables of the genetic algorithm to find are limited to the resistance R� that determines the amplification ratio of the amplifier output current, resistance R�, resistance R� which determines the dynamic characteristics of the linear amplifier current, and condenser C�. At this point, if the amplification ratio of the amplifier output current is given, one less genetic algorithm variable needs to be found as the resistances R� and R� have a proportional

Next, the searching range of the parameters to be identified is limited according to the characteristics of each device. As the resistance R� which determines the amplification ratio of the amplifier output current is a signal resistance, it is desirable to have a high resistance value. Therefore, in the case of resistance R� which determines the amplification ratio of the amplifier output current, it has to be sought in the kΩ range. In contrast, resistance R� which determines the dynamic characteristics of the linear amplifier circuit has to be sought in a wide range. For condenser C�, which determines the dynamic characteristics of the linear amplifier circuit, a value in the nF to μF range is ideal when considering the dynamic

After that, the objective function is determined to implement the genetic algorithm. The objective in this program is the design of a linear current amplifier that has a current output in the form that the designer seeks. Therefore, it has the form shown in Equation (23) and the response of the system that satisfies the time response characteristics defined by the

G��s� � � ������

At this point, the randomly given d�, d�, e� and e� are the coefficients of the system G� that satisfies the time response characteristics. The objective function to implement the genetic

Here, e�t� � g��t� � g����t�, g��t� is the step response of the system defined by the designer

Finally, the parameters to operate the genetic algorithm, such as the size of the entity group, the maximum chromosome length, maximum number of generations, crossbreeding probability, and mutation probability, are defined. Here, in order to improve the performance of the implemented genetic algorithm, configuration for methods such as the

and g����t� is the step response of the current amplifier transfer function.

penalty strategy, elite strategy, and scale fitting method is necessary.

���������

(25)

F��� � � e�t��dt (26)

following manner utilizing genetic algorithm.

characteristics of the current amplifier.

designer is defined as shown in Equation (25).

algorithm is defined as shown in Equation (26).

relationship.

**Figure 9.** Genetic algorithm

Genetic algorithms can be categorized into BCGA(Binary Coded Genetic Algorithm), SGA(Signal Genetic Algorithm), and RCGA(Real Coded Genetic Algorithm) depending on the expression of the chromosome. Generally, RCGA is used for optimization problems regarding continuous search domain variable with constraints. This is because if the chromosome is expressed by real code, genes that match perfectly with the variable in question could be used and the degree of precision of the calculation is only dependent on the calculation ability of the computer regardless of the length of the gene.
