*2.2. Proposed Method*

The equations of normal force and levitation force used in the control of a maglev train can be obtained through FEM analysis [17] and the structural properties of the target vehicle. Equations (4) and (5) show the relationship between normal force and slip frequency, and driving force and slip frequency, respectively:

$$F\_N = \frac{l\tau\mu\_0}{2} \frac{1 - \left(R\_m S\right)^2}{\left(\sinh\theta\,\mathrm{g}\right)^2 + \left(R\_m S\cosh\theta\,\mathrm{g}\right)^2} \left(Z\_m I\_m\right)^2\tag{4}$$

$$F\_T = l\tau\mu\_0 \frac{R\_mS}{\left(\sinh\theta\,\mathrm{g}\right)^2 + \left(R\_mS\cosh\theta\,\mathrm{g}\right)^2} (Z\_mI\_m)^2\tag{5}$$

where *Rm* = σ*t*μ0λ*f* is the magnetic Reynolds number, σ*<sup>t</sup>* = σ*teff* is the effective conductivity, *teff* is the effective thickness of the secondary conductor, β = <sup>π</sup> <sup>τ</sup> is the air:gap:wavelength ratio, *<sup>S</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *Vm Vsy* <sup>=</sup> *fsl f* is the slip, *Vm* is a variable representing the speed of the train, *Vsy* is a variable representing synchronous speed, *Zm* is maximal winding density (the maximal winding density per unit length of the train core), *Im* is the maximal current (the maximal current input for the required current to operate the system), *g* is the effective void, τ is the pole spacing, *f* is the power frequency, σ is the conductivity of the secondary conductor, *fsl* is the slip frequency, and *l* is the primary core width.

By deriving the relational expression between driving force and normal force from the relational expressions in Equations (4) and (5), the ratio of normal force and driving force, as shown in Equation (6), can be derived using the equation related to slip frequency:

$$\frac{F\_N}{F\_T} = -\frac{1}{2}(R\_m S - \frac{1}{R\_m S})\tag{6}$$

where *Rm* <sup>=</sup> <sup>σ</sup>*t*μ0λ*<sup>f</sup>* and *<sup>S</sup>* <sup>=</sup> *fsl <sup>f</sup>* . From this result, *RmS* is slip frequency.

If this is summarized in terms of slip frequency, it can be expressed as Equations (7) and (8). Through Equation (8), the maximal normal force can be obtained from the currently running fixed slip-frequency value. If this maximal normal force is then substituted for the required thrust, the maximal usable slip frequency range can be calculated for each operating condition. That is, if the normal force and the propulsion force at the maximal propulsion force of a currently running train are substituted, slip frequency can be determined and controlled so as to be limited to a range that does not fail in levitation.

$$0 = (\sigma\_t \mu\_0 \lambda)^2 f\_{sl}{}^2 + 2 \frac{F\_\mathcal{N}}{F\_T} (\sigma\_t \mu\_0 \lambda) f\_{sl} - 1 \tag{7}$$

$$f\_{sl} = \frac{-2\frac{F\_{\rm IN}}{F\_T}\sigma\_l\mu\_0\lambda \pm \sqrt{\left(2\frac{F\_{\rm IN}}{F\_T}\sigma\_l\mu\_0\lambda\right)^2 + 4\left(\sigma\_l\mu\_0\lambda\right)^2}}{2\left(\sigma\_l\mu\_0\lambda\right)^2} \tag{8}$$

Figure 4 shows the ratio of normal force/thrust force according to slip frequency using Equation (6). Table 2 uses Figure 4 and Equation (8) to calculate the slip frequency in which the maximal normal force is generated within the range of levitation not failing when propulsion force is changed (if slip

frequency is negative, it operates as a braking mode; if it is positive, it operates as a powering mode for thrust).

**Figure 4.** Ratio of normal force and thrust force according to slip frequency.


**Table 2.** Normal force margin ratio by slip frequency at each thrust.

Table 2 shows that, when the maximal driving force was 100%, when the driving force decreased, the normal force also decreased. Therefore, when using the same slip frequency of 13.5 Hz at approximately 75%, 50%, and 30% thrust, the amount of normal force generated on the basis of maximal thrust is reduced. Assuming that the margin ratio of the normal force is 1 at a slip frequency of 13.5 Hz, margin rates at each operating condition are 1.33, 2, and 3.33, respectively. If slip frequency was lowered by this margin factor, efficient operation would be possible within the range of normal force that did not affect safety. If the optimal slip frequency suitable for train operation conditions were derived in this way, a train-operation pattern capable of varying the appropriate slip frequency during train operation could be obtained.
