CC BY
1УДК 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:
s 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.
s The conductivity of PM leakage fluxes in the magnetic system of the generator remains unchanged.
s The magnetic resistance of the ferromagnetic details is ignored. s The generation of eddy currents and their influence on the magnetic parameters are neglected.
s 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 51 and S2, mm,
- FS rotation speed, ®, turnis,
- 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]).
2. Referencing the directory, the following parameters of the PM material are registered:
- the coercive force by induction, HcB, Aim,
- the residual induction, Br, T,
- the maximum energy product, (BH)m, Jim3.
The specified quantities are given by an interval or a condition, so they must be specified using the following conditions [2]:
CB, r <(BH)m<HcBBr,
Br «2^ (BH)
B
1Г1) 2
(BH)
___r
m ~
4Цо'
(BH)m .a (BH)m
—-<Hcb <4—-,
Br Br
where p.0 =4n10-7Hn/m. 3. 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:
ko
0
Bo = ^--
2Yc
ko 2- 4^o ycmpHcBBr
where k0 =(Br+^0 mpHcB). Determination of magnetic field strength between poles of the PM:
Ho = .
0 -omp
Determination of the convexity coefficient of the PM material:
Y _ 2 /HcBBr _ HcBBr Yc " ^ (BH)m (BH)m .
The permeability coefficient of a cylindrical PM with a ratio ^^ in the range
lPM
of 0.1^10 is determined by the following expression:
mp _2'46 (dPMM)-1'32,
when dpM _ 1, mp _ 3.
lPM F
Determination of magnetic induction at the N pole PM (Bn=Bs):
bn = (цо Ho + Bo)"
lPM
4lPM2+dPM
Determination of the magnetic induction of leakage of PM:
bg = Bo-Bn.
The operating point Ao on the B(-H) characteristic of the PM is shown in Fig. 1.
iB
br
b0 bn
bN
bN b0
bN
0
h0"
Fig. 1. B(-H) characteristic and working points of the PM.
4. After installing the PM in the magnetic system of the generator, the operating point along the return line moves to positions at Si, and positions A0 at (Fig. 1). At the corresponding points the magnetic parameters are determined as
H'(") -Ho -
BoHcf
Bo-ki(Ho-Hcf)'
Bo( ) - kiHo( ), Pr - ÏT (! - Yc),
1 HcB c
Hcf - B0 + H0, pr
k = ÎPM A k1 ç AS ,
where the pr is the return coefficient, Hcf is the fictitious coercive force
BN - B0 -ba.
Determination of the magnetic conductivity of air gaps S1 and S2 (Fig. 2 [1])
SPM
Л,
- Mo [(^
) + 20+75^ + 0'48d™
^0
where SPM is the area of PM's transverse cross-section.
5. Determining value of the change in the average magnetic flux AOpMav along the length of the PM in the extreme positions of the FS (Fig. 3, 4 [1])
AOpMav = ®''PMav-®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,
O' 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, O' 'PMav- B'' PMavSPM.
6. Determining the magnitude of the change in magnetic flux AOEMav in the electromagnet core at the extreme positions of the FS
AO em= 0''em-o'em. Determining the value of the magnetic flux in the absence of a FS
O' EM -BN Sfc.
Determining magnitude of the magnetic flux in the presence of a FS
0''em=0.
7. Determining the duration of the formation of one half-cycle of the EMF in the IC
At _ 1
2mw
8. The relationship between the number of turns of IC and the amplitude value of the EMF
w _ einAt wic—-■
AO
9. 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
