# Reducing the Power Consumption of the Electrodynamic Suspension Levitation System by Changing the Span of the Horizontal Magnet in the Halbach Array

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## Abstract

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## 1. Introduction

## 2. Analytical Methods

_{r}is PM’s remanence, h

_{m}is PM’s thickness, M is the number of permanent magnets per wavelength of the Halbach array and k is calculated as

_{hm}is the length of the horizontally magnetized PM and l

_{vm}is the length of the vertically magnetized PM.

_{hm}. The geometrical, material and dynamic parameters used in the simulations are described in Table 1. At first, we used a magnetostatic solver, which allowed us to calculate the waveforms of magnetic induction in the air gap for three different cases of fill factor. Second, we used a transient solver. The analysis investigated a Halbach array magnets arrangement moving along an aluminum plate. The virtual forces were read from the whole permanent magnets’ set. The Halbach array consisted of 4 wavelengths in order to reduce the end effects. The simulations were conducted for six different velocities of the PM array to observe the skin-effect.

## 3. Results

#### 3.1. Magnetic Field Waveform

_{hm}= 0.2, 0.5 and 0.85. A fill factor of 0.2 means that only 20% of the wavelength λ is filled with horizontally magnetized PM, while 0.5 refers to a classic Halbach array. The cases where fill factors are equal to 0.2 and 0.85 are significant, because, for these values, the power consumption reaches the local minimum, which is shown later in the paper. Yellow and green blocks correspond to horizontally magnetized permanent magnets, while red and blue blocks correspond to vertically magnetized permanent magnets.

_{hm}= 0.2 creates a magnetic field; the B

_{x}component is triangular, narrow and has the highest peak value, while the distribution of the B

_{y}component is trapezoidal with the lowest maximum value. In the Halbach array with γ

_{hm}= 0.85, the components have opposite waveforms. The B

_{x}waveform is trapezoidal, while the B

_{y}component is triangular and has the highest peak value. The average values of the magnetic field magnitudes B

_{m}are calculated in Table 2. The classic Halbach array with γ

_{hm}= 0.5 has the highest average value, while the arrangement with low horizontally magnetized PM γ

_{hm}= 0.2 has the lowest value.

#### 3.2. Transient Simulation Results

_{max}= 100 m/s.

_{hm}= 0.3, while the braking force (Figure 5b) is the lowest when γ

_{hm}= 0.9. That means that the most effective, in terms of the mass of the PMs, is the arrangement with a fill factor of 0.3 and the arrangement causing lowest drag forces is the one with a fill factor of 0.9. To compare the cases in terms of power consumption, we conducted further analyses.

#### 3.3. Calculation of the Required Levitation Force

_{body}= 14,000 kg. The system was designed to begin the levitation at velocity v

_{lew}= 10 m/s. This means that, at this velocity, the created levitation force is equal to the total weight of the vehicle Q

_{tot}, which consists of the body car weight Q

_{body}and the levitation magnets weight Q

_{PM}, as in

_{hm}= 0.9 creates the lowest levitation force, so the permanent magnet mass is higher than the mass of an array with γ

_{hm}= 0.3, which creates the highest levitation force. The permanent magnets mass for a given fill factor can be calculated with

_{tot}. The results are shown in Figure 6.

#### 3.4. Power Consumption Analysis

_{hm}= 0.5 is 1.15 times higher than that at γ

_{hm}= 0.2. When it comes to absolute values, the power saving for the case under consideration equals to more than 0.156 MW, as it is shown in Table 3.

## 4. General Discussions

_{hm}= 0.5) is used. Here, we show that, for the system described in Table 1 (with the γ

_{hm}= 0.2), the power savings at velocity = 100 m/s can even reach 15% and the higher the velocity, the greater the percentage of power-saving. The fill factor described in this paper is the best for a given EDS system. For other geometries of the magnetic suspension system, the value of the best fill factor may slightly differ.

## 5. Conclusions

_{hm}= 0.5). Such an approach changes the waveform of magnetic induction in the airgap and, as a result, changes the forces created by the traveling PM array. Although the overall volume (and mass) of the PM arrangement may remain constant, selecting the appropriate γ

_{hm}factor allows researchers to design a better power-saving levitation system capable of carrying the weight of the traveling high-speed magnetic railway. We conducted a series of numerical computations with the usage of Ansys Maxwell. Based on the simulation results, further calculations were made. It was calculated that the arrangement generating the lowest instantaneous power is the Halbach array with fill factor γ

_{hm}= 0.2 for each considered velocity. This value was determined with an accuracy error of 5%.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Halbach array with a wide vertically magnetized permanent magnet over a conductive track. Side view in picture (

**a**) and front view in picture (

**b**).

**Figure 2.**Examples of Halbach arrays with different fill factors. (

**a**) γ

_{hm}= 0.2, (

**b**) γ

_{hm}= 0.5 and (

**c**) γ

_{hm}= 0.85.

**Figure 3.**A section of the magnetic field in the air gap beneath the Halbach arrays. (

**a**) Magnitude of magnetic induction, (

**b**) x component of magnetic induction and (

**c**) y component of magnetic induction.

**Figure 4.**Results obtained from transient simulations. (

**a**) Levitation force as a function of velocity and fill factor, (

**b**) braking force as a function of velocity and fill factor, (

**c**) lift-to-drag ratio as a function of velocity and fill factor and (

**d**) lift-to-weight as a function of velocity and fill factor.

**Figure 5.**Results obtained from transient simulations for v

_{max}= 100 m/s. (

**a**) Levitation force as a function of fill factor, (

**b**) braking force as a function of fill factor, (

**c**) lift-to-drag ratio as a function of fill factor and (

**d**) lift-to-weight as a function fill factor.

**Figure 6.**Weights of vehicle body (yellow line), permanent magnets (red line) and sum of both (blue line). Weight of PM is calculated for v

_{lew}= 10 m/s.

**Figure 7.**(

**a**) Rescaled levitation force as a function of velocity and fill factor and (

**b**) rescaled braking force as a function of velocity and fill factor.

**Figure 8.**Instantaneous power required to overcome the braking forces. Power consumption as a function of fill factor and velocity.

**Figure 9.**Instantaneous power required to overcome the magnetic braking forces and its components (Equation (11)) at v = 100 m/s.

**Figure 10.**The difference of required power for fill factor = 0.5 and fill factor = 0.2 as a function of velocity.

Parameters Type | Parameter | Symbol | Value |
---|---|---|---|

Geometrical parameters | Wavelength | λ | 400 mm |

PM’s height | h_{m} | 50 mm | |

Conductive plate height | h_{p} | 10 mm | |

Air gap size | g | 30 mm | |

PM’s width | w_{m} | 50 mm | |

Conductive plate width | w_{p} | 120 mm | |

Material parameters | Plate conductivity | s | 32.47 MS/m |

PM’s remanence | Br | 1.4 T | |

Dynamic parameters | Max vehicle velocity | v_{max} | 100 m/s |

Levitation velocity | v_{lev} | 10 m/s | |

Mass of the vehicles body | m_{body} | 14,000 kg |

**Table 2.**Comparison of average vales of magnitude of magnetic induction for different cases of fill factor.

γ_{hm} = 0.2 | γ_{hm} = 0.5 | γ_{hm} = 0.85 | |
---|---|---|---|

B_{m}_{_avg} | 190 mT | 230 mT | 213 mT |

**Table 3.**Comparison of ratio and absolute difference of required power for vehicle traveling with velocity = 100 m/s.

P_{γhm} _{= 0.5}/P_{γhm} _{= 0.2} | P_{γhm} _{= 0.5} − P_{γhm} _{= 0.2} |
---|---|

115% | 0.156 MW |

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**MDPI and ACS Style**

Kublin, T.; Grzesiak, L.; Radziszewski, P.; Nikoniuk, M.; Ordyszewski, Ł.
Reducing the Power Consumption of the Electrodynamic Suspension Levitation System by Changing the Span of the Horizontal Magnet in the Halbach Array. *Energies* **2021**, *14*, 6549.
https://doi.org/10.3390/en14206549

**AMA Style**

Kublin T, Grzesiak L, Radziszewski P, Nikoniuk M, Ordyszewski Ł.
Reducing the Power Consumption of the Electrodynamic Suspension Levitation System by Changing the Span of the Horizontal Magnet in the Halbach Array. *Energies*. 2021; 14(20):6549.
https://doi.org/10.3390/en14206549

**Chicago/Turabian Style**

Kublin, Tomasz, Lech Grzesiak, Paweł Radziszewski, Marcin Nikoniuk, and Łukasz Ordyszewski.
2021. "Reducing the Power Consumption of the Electrodynamic Suspension Levitation System by Changing the Span of the Horizontal Magnet in the Halbach Array" *Energies* 14, no. 20: 6549.
https://doi.org/10.3390/en14206549