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F
mr
A A A
m1
F
m
mr
m1
F
0,
(4)
where the input of the element into the functional is in matrix form:
F
m
A
1
36V
m
m Vm
N
A
,
4
(5)
where:
V m – is the size of tetrahedron m;
[ N m ] – the matrix of tetrahedrons;
{ } – the vector of the current density components in tetrahedron;
r - the number of tetrahedron of the calculations space.
On the basis of expression (4) we can get the system of non-linear algebraic equations the solution
of which provides functional minimum (3) giving thus the solution of the equations system (2).
For the analysis of the magnetic field calculations the value of magnetic induction is applied
represented with the following equation by means of vector magnetic potential [10]:
B = rot A . (6)
The analysis of magnetic field is based on the components of magnetic induction represented as
follows [8]:
B
B
B
x
y
z
A
A
z y
,
(7)
y
z
Ax
Az
,
(8)
z
x
Ay
Ax
.
x
y
(9)
The description of the realization of the method of finite elements (MFE) for the calculation of
three-dimension magnetic field of electric machine and the program are represented in [9, 10].
Calculation results
The three-dimension magnetic field in electrical AC machine is analysed by means of induction
machine model demonstrated in Figure 1. The calculations and analysis are made for the machine with
symmetric air gap (without defects) and different levels of non-symmetry of it (with the defects in bearing
units, e.g. misalignment of rotor, tapering and ellipse form of rotor, eccentricity of the stator bore).
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