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Hovhannisyan A.T.

  


THE MATHEMATICAL APPARATUS FOR CALCULATING THE MAGNETIC SYSTEM OF AN «PERMANENT MAGNET-ELECTROMAGNET» INDUCTION GENERATOR (2) *

  


Аннотация:
this paper presents a mathematical apparatus for calculating the parameters of a “permanent magnet-electromagnet” induction generator. The study focuses on the extreme positions of ferromagnetic shunts within the magnetic system. Analytical expressions are derived for determining the magnetic parameters at the operating point of the permanent magnet and the corresponding variations in magnetic flux within the induction coils. The paper also explores the relationships between the electromotive force induced in the induction coils, the number of permanent magnets, the rotational frequency of the ferromagnetic shunts, and the number of turns in the induction coils. The findings are applicable to the design and optimization of permanent magnet systems, particularly in the context of induction generators   

Ключевые слова:
induction, generator, permanent magnet, electromagnet, coil, operating point   


DOI 10.24412/2712-8849-2024-978-337-342

УДК 621.318.2

 Hovhannisyan A.T.

Candidate of Technical Sciences, Leading Researcher

National Polytechnic University of Armenia

(Yerevan, Armenia)

 

THE MATHEMATICAL APPARATUS FOR CALCULATING 

THE MAGNETIC SYSTEM OF AN «PERMANENT 

MAGNET-ELECTROMAGNET» INDUCTION GENERATOR (2)

 

Abstract: this paper presents a mathematical apparatus for calculating the parameters of a “permanent magnet-electromagnet” induction generator. The study focuses on the extreme positions of ferromagnetic shunts within the magnetic system. Analytical expressions are derived for determining the magnetic parameters at the operating point of the permanent magnet and the corresponding variations in magnetic flux within the induction coils. The paper also explores the relationships between the electromotive force induced in the induction coils, the number of permanent magnets, the rotational frequency of the ferromagnetic shunts, and the number of turns in the induction coils. The findings are applicable to the design and optimization of permanent magnet systems, particularly in the context of induction generators.

 

Keywords: induction, generator, permanent magnet, electromagnet, coil, operating point.

 

Introduction. Structure, operation principles, main dynamic characteristics, electrical circuits for replacing the magnetic core, features of the magnetic system, and other details of an induction generator with a “permanent magnet-electromagnet” (PMEIG) system as a source of electricity are presented in [1].

This paper presents the mathematical apparatus for calculating PMEIG.

The Problem.

Develop a mathematical apparatus for calculating PMEIG, that includes:

  • determination of the magnetic parameters of the position of the permanent magnet (PM) operating point depending on the extreme position of the ferromagnetic shunts (FS),
  • determination of the magnitude of change in magnetic flux in induction coils (IC) depending on changes in the extreme positions of the FS,
  • identify the correlation between the magnitude of the induced electromotive force (EMF) in the IC, the number of PMs, the rotation frequency of the FS, and the number of turns of the IC.

The Solution

In the process of developing the mathematical apparatus, the following permissions were made:

  • The material of the PM is homogeneous and undergoes magnetization in a homogeneous magnetic field along the corresponding axis until reaching saturation, as well as get stabilization.
  • The conductivity of PM leakage fluxes in the magnetic system of the generator remains unchanged.
  • The magnetic resistance of the ferromagnetic details is ignored.
  • The generation of eddy currents and their influence on the magnetic parameters are neglected.
  • The presence of a short circuit in the IC is excluded.
  1. Preliminary data was as follows:

- the brand of the PM,

- the length of the PM, ℓPM, mm,

- the diameter of the PM, dPM, mm,

- working air gaps δ1 and δ2, mm,

- FS rotation speed, ω, turn/s,

- permanent magnet number, m, pc.,

- number of turns of IC, wI, w2, or the required amplitude value of EMF, einPM, einEM, V (Fig. 3, 4 [1]).

  1. Referencing the directory, the following parameters of the PM material are registered:

- the coercive force by induction, , A/m,

- the residual induction, Br, T,

- the maximum energy product, (BH)m, J/m3.

The specified quantities are given by an interval or a condition, so they must be specified using the following conditions [2]:

 

 

where =4π10-7Hn/m.

  1. Determination of the magnetic parameters of the PM magnetized outside the magnetic system [3, 4].

Determination of the magnetic induction in the 00 neutral part of the PM:

 

where=(Br+HcB).

Determination of magnetic field strength between poles of the PM:

 

Determination of the convexity coefficient of the PM material:

  

The permeability coefficient of a cylindrical PM with a ratio  in the range of 0.1÷10 is determined by the following expression:

 

 

Determination of magnetic induction at the N pole PM (BN=BS):

 

Determination of the magnetic induction of leakage of PM:

 

The operating point A0 on the B(-H) characteristic of the PM is shown in Fig. 1.

Fig. 1. B(-H) characteristic and working points of the PM.

 

  1. After installing the PM in the magnetic system of the generator, the operating point along the return line moves to positions at δ1, and positions at δ2 (Fig. 1).

At the corresponding points the magnetic parameters are determined as

 

 

where the  is the return coefficient,  is the fictitious coercive force

 

Determination of the magnetic conductivity of air gaps δ1 and δ2 (Fig. 2 [1])

 

where SPM is the area of PM’s transverse cross-section.

  1. Determining value of the change in the average magnetic flux ΔΦPMav along the length of the PM in the extreme positions of the FS (Fig. 3, 4 [1])

ΔΦPMav = Φ′′PMav−Φ′PMav.

Determining the average values of magnetic flux along the length of the PM in the absence of a FS

B′PMav ≈(B′0+B′N)/2,

Φ′PMav =B′ PMavSPM.

Determining the average values of magnetic flux along the length of the PM in the presence of a FS

B′′ PMav =(B′′0+B′′N)/2,

F¢¢PMav= B′′ PMavSPM.

  1. Determining the magnitude of the change in magnetic flux ΔΦEMav in the electromagnet core at the extreme positions of the FS

DFEM= F¢¢EM-F¢EM.

Determining the value of the magnetic flux in the absence of a FS

F¢EM =SFC.

Determining magnitude of the magnetic flux in the presence of a FS

Φ′′EM=0.

  1. Determining the duration of the formation of one half-cycle of the EMF in the IC

 

  1. The relationship between the number of turns of IC and the amplitude value of the EMF

wIC

  1. From relevant literature sources the remaining parameters of the IC are selected and calculated: wire grade, wire diameter, coil window, etc.

The coil windings can be connected in various combinations, such as series, parallel, etc., depending on the magnitude, power and nature of the output voltage.

 

 

Conclusion.

The mathematical apparatus developed for the magnetic system of the PMEIG can serve as a basis for the calculation and design processes of systems utilizing PMs, particularly in the context of induction generators.

Gratitude.

The work was carried out in the main scientific laboratory “Automation and Electromagnetic Systems”, funded by the Ministry of Education, Science, Culture and Sports of the Republic of Armenia.

 

REFERENCES:

 

  1. Hovhannisyan A.T. The structure and operation principle of a permanent magnet-electromagnet of an induction generator (1) // Инновационные научные исследования. 2023. № 6-2(30). C. 55-61;
  2. Hovhannisyan A.T., Yenokyan K.R. Approaches to the selection of initial magnetic parameters for the calculation of a permanent magnet and evaluation of calculation websites // Инновационные научные исследования. 2023. № 8-1(31). C. 34-40;
  3. Hovhannisyan A.T. The mathematical apparatus for calculating the scattering coefficient of a cylindrical permanent magnet magnetized along the length // Инновационные научные исследования. 2022. № 5-2(19). C. 58-64;
  4. Hovhannisyan A.T. Mathematical apparatus for calculating a cylindrical permanent magnet magnetized along the length based on usecase // Инновационные научные исследования. 2023. № 6-2(30). C. 62-72
  


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Ссылка для цитирования:

Hovhannisyan A.T. THE MATHEMATICAL APPARATUS FOR CALCULATING THE MAGNETIC SYSTEM OF AN «PERMANENT MAGNET-ELECTROMAGNET» INDUCTION GENERATOR (2) // Вестник науки №9 (78) том 2. С. 337 - 342. 2024 г. ISSN 2712-8849 // Электронный ресурс: https://www.вестник-науки.рф/article/17183 (дата обращения: 27.03.2025 г.)


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