5. energy conversion
5.6
Week 13
Magnetic field energy and forces
In order to be able to analyise mathematically the electromechnical system that is completely
described by the energy balance equation, we need to be able to determine qualitatively the
energy of the magnetic field and the associated force.
5.6.1 Magnetic field
A magnetic field is a region of space in which certain physical effects occurs in particular
the development of mechanical force. A pictorial model of the field can be made by drawing
closed loops of magnetic flux, such that their direction and spacing at any point are a
measure of the flux density. The magnetic circuit in the present context is composed partly
of ferromagnetic material such as iron, and partly of an airgap. The iron serves to “guide” the
flux in a desired path; the
airgap is necessary to make useful magnetic effects readily
accessible.
The lines in a flux plot have no real existence. In a given region a magnetic field may change
direction, become weaker in some place and stronger in others.
5.6.2 Magnetic circuit n/a
Engineers look upon magnetic flux (Weber) as produced by electric current. A current I
develops around any path that links it a magneto motive force (M.M.F) F = I (ampere). The
effect of a current can be multiplied. By coiling the electric circuit into N turns so that
around a path linking all N turns the m.m.f is N times as great, giving F =- ampere- turn.
The m.m.f is distributed along the path, to give along a path element of length dx the
magnetic field intensity h (ampere-turn/ metre). The summation of Hdx around a single loop
closed with F i.e F = Hdx = m.m.d.
5. energy conversion
Week 13
At any point, H gives rise to a flux density B = NH (tesla or Weber/m2) our Henry/meter]
Flux summation of the flux density over the area available to the flux path given the total
flux [ i.e Ø i.e. BA. (Weber).-2.6 where A is the are of flux path.
The ‘ law of the magnetic circuit relates the total flux Ø to
the mmf f through the
expression.
Ø = F =F Comparable to the law of electric circuit 2.7
S
I
=
Where s
V
v.g (ohm’s law)
R
=
=
the total reluctance (ampere-turn per weber]
And = 1/s = total permeance [ weber per ampere-turn]
For a path- length x of materials of absolute permeability U, and having a uniform crosssectional area A over which the density B is everywhere the same, the mmf f require = N x x
= Hx…………..2.8
From equation 2.6, 2.7 and 2.8, F Øs F = mmf
And the reluctance of the path S =
f/Ø = Hx
2.9
= x……….2.10
And the 1/s UA/x
For a succession of parts , x, y, z
F = fx + fy + f2 +
and S = SX + SY + SZ +
2.11
2.12
If, however the parts are in parallel and share the flux
F = fx =fy =fz and
For fields in ferromagnetic materials U is very much greater, and the relative permeability Nr
=u =Uo 4 /107 1/80000
5. energy conversion
Week 13
Which means that H = Ub = 800000B
For field ferromagnetic materials U is very much greater,
Ad the relative permeability Ur = U
Uo
Since, usually Ur is large, then it is convenient ( it simplies analysis) to assume that the
whole mmf is required for the excitation of the air gap i.e the whole of the field energy is
stored in the air-gap
5.7
Magnetic field energy
With the assumption that the magnetic filed energy is concentrated within the air gap. It
becomes easy to calculated the magnetic field energy.
A magnet attract on iron bar. If the iron bar is light enough and the magnet filed is enough,
the bar will be seen to move up to get attached to the magnet. The movement of the bar
signifies that work is done, since the iron bar has mass and covered some distance
(work done = force x distance). This means that the space that the file occupies (the field
region) can demonstrated or has on attribute of force. And hence, the filed region must
process some energy. If can be easily noticed that the force is strong when the air gap is short
but rapidly diminishes as the air gap length is increased.
5.8 Maxwell stress
Fig 2.5 maxwell forces
Maxwell formulated the concept that the forces is transmitted across the gap between a pair
of magnetized surface as a result of two stresses. If at a point in the gap the flux density is B
and the corresponding field intensity is H =B/U,. then there is a tensile stress of magnitude
5. energy conversion
Week 13
1.2 BH along the direction of a flux line and a compressive stress ½ BH along all directions
at right angles to a flux line.
Fig 2.5 shows two iron bars forming part of magnetic circuit when, as at (a), the polar
surfaces are close together, the flux is mainly concentrated between the surfaces. The density
B is large, and so therefore is H, and ½ BH represented a strong tensile force of attraction
between the faces. Not all the flux is useful; some, of the leakage flux, exists at the sides of
each bar. Flux crossing the boundary between air and a high permeable materials must enter
or leave the boundary between air and a high permeable materials must enter or leave the
boundary almost at right angles, so that the tensile stress due to faces. All the comprehensive
stresses balance out by symmetry.
In case (b), the greater reluctance of the long air gap reduces the total flux, the useful flux
density of the pole faces is smaller while the leakage flux is much greater hence the forces of
attraction between the pole faces is much less than in case (a)
In most practical applications, the air gap is small enough to enable us assumes a uniform
flux density over the polar area. i.e in the air gap.
.