UNIT-2
MAGNETIC CIRCUITS
Estimation of total MMF
The estimation of total mmf required to establish magnetization
in the magnetic circuit involves the knowledge of dimension and
configuration of the magnetic circuit. The magnetic circuit is
split in to two convenient parts which may be connected in series
or parallel. The flux density is calculated in every part and mmf
per unit length is found by B-H curve of the concern material.
The parts of magnetic circuits are:
i. Air gap
ii. Core: Stator core and rotor core
iii. Teeth: Stator teeth and rotor teeth
iv. Pole : In case of dc machine & salient pole synchronous m/c
v. Yoke: In case of dc machine
MMF for Air-Gap
The iron surfaces around the air gap are not smooth due to one or
both of iron surfaces around the air gap may be slotted or radial
ventilating ducts in the machine for cooling or presence of
salient poles in the machine, so the calculation of mmf by
ordinary methods give wrong results. These problems are solved
with special techniques.
Let
Case-1
Consider the iron surfaces on the two sides of the air gap to be
smooth and the gap length to be uniform.
Let
Case-2
Consider the iron surfaces on one side of the air gap to be
smooth and slotted on the other side. Assume flux path through
air gap for one slot pitch is confined to width wt.
As shown in figure flux can not confine to a width alone but to
fringing it will spread over to a greater width. The value of
reluctance now decrease but more than in case-1.
Consider that in a slot pitch the flux is spread over to a width ys’.
or
kc = carter’s coefficient which is the function of the ratio of Slot
opening to the gap length.
The effect of slotting can be considered in two ways
i)The slot pitch ys is contracted so changing the air gap flux
density B’.
Where
ii) The air gap length lg is enlarged to lg’
where
Case-3
Consider the iron surfaces on both sides of the air gap to be
slotted.
As per case-2
Here kc1 and kc2 are carter’s coefficients for stator and rotor slots
respectively and are the function of (wo1/lg1) and (wo2/lg2).
Case-4
Consider the iron surface of the air gap having radial ventilating
ducts.
As per case-2 and 3
a) With number of ducts on one side, the core length will be
(L-kd.nd.wd) and change in flux density from B to B’;
There is increase in gap density by factor kgd.
kd is the carter’s coefficient which is function of (wd/lg)
b) Consider ducts on both sides facing each other;
where
Here kd1 and kd2 are carter coefficients for stator and rotor ducts
and function of [wd1/(1/2.lg)] and [wd2/(1/2.lg)] respectively.
Case-5
Consider the effect of saliency. In salient poles the air gap is
minimum at the centre along the direct axis and increases as the
pole tip is approached. In this case the mmf for air gap per pole
for a machine with salient pole on one side
and smooth surface on the other is:
lg= minimum gap length at centre
Bg= peak value of flux density
kf = field form factor
= Bav/Bg
= pole arc/pole pitch
MMF for Teeth
Two problems arise when calculating the mmf for teeth:
i) The slots of all the machines are parallel sides which make
the teeth tapered. So the area through which the flux is
passing in the teeth is not constant and giving different values
of flux density over the length of the teeth.
ii) The teeth are working at high flux density, so mmf should
high enough to make an appreciable flux passes through the
slots and ducts too. This make the calculation complicated.
Estimation of mmf for tapered teeth: three methods are usually
adopted:a) Graphical method
b) Three ordinate method
c) One third density method
Graphical Method
Due to tapering of the tooth, the flux density at each section is
different. Calculate density at a large number of sections spread
over the entire length of the tooth and find the corresponding
intensity (h) or ampere turn per meter (at/m) from the B-H curve
of material. Plot the graph showing ‘h’ varies over the length of
the tooth. The mean ordinate of this graph gives the equivalent
‘h’, call hmean.
Mmf for tooth = ATt = hmean * lt
Where lt = height of tooth
Three Ordinate Method
This method is applicable to simple trapezoidal teeth of moderate
taper.
Let h1, h2, and h3 be the at/m for three equidistance sections of the
taper. Applying Simpson’s rule
hmean= 1/6 (h1+4h2+h3)
Mmf for tooth = ATt = hmean * lt
Where lt = height of tooth
One Third Density Method
This method is applicable for slight taper and low densities. It is
based on the assumption that the average value of tooth flux
density is equal to the density at a section 1/3rd of the tooth
height from the narrower end.
Let (Bt)1/3 = flux density at 1/3rd tooth height from narrower end
(ht)1/3= corresponding at/m from the B-H curve
Mmf for tooth = ATt = (ht)1/3* lt
Where lt = height of tooth
Real and Apparent Flux Densities
For high value of tooth flux density the mmf should high enough
to pass an appreciable flux through the slot and ducts. Therefore
calculation of density through the tooth area alone is not correct.
This flux density is called apparent flux density Bapp. The actual
flux passing through the tooth is always less than the total flux.
So real flux density is always less than apparent flux density.
Let
Φs = total flux over one slot
Φi = flux passing through tooth alone
Φa = flux passes through slot alone
Φs = Φi + Φa
Divide the above equation by Ai teeth area over a slot pitch
Where Aa = area facing the flux ϕa
= ysL – wt Li
Φa/ Aa = Ba = µo h = 4π * 10-7 h
where h is the AT/m for real tooth density B real.
Further
Electromagnets
Electromagnets have wide variety of applications in the field
of pulling, lifting and holding. They are used in magnetic
switches and relay, circuit breakers, lifting industrial loads of
scraps, steel plates heavy castings etc.
The main area of design deals with force and ampere turn
calculations, temperature rise and heat dissipation.
Magnetic pull or Force:
F = force between two poles, N
B = flux density in air gap, wb/m2
A = cross-sectional area of each pole facing air gap, m2
Let upper pole is pulled by a distance dx,
work done = F *dx Joule
Work done creates a change in stored magnetic energy.
Change in stored magnetic energy = (energy stored/m3) * change
in volume
From above equations
The ampere turns requirement:
The ampere turn calculated by knowing the flux density and
finding at/m from B-H curve of the material.
Ati = ampere turn for iron parts
Atg = ampere turn for air gap
ATg = 800,000 Bg * lg
Total ampere turn AT = Ati + Atg
normally Ati = 20% to 30% of Atg
Temperature rise:
Total losses or heat developed in exciting coil of an
electromagnet is
W = I2R
Where W = heat developed in the coil, W
I = current in the coil, A
R = resistance of the coil, ohms
Let
C = cooling coefficient, ˚C-m2/W
θm= final temp. rise of cooling surface, ˚C
The heat dissipating surface of the coil is considered to be only
the inner and outer surfaces,
s = 2lmt*h
Where lmt = length of mean turn
h = height of the winding
thus
let
θm= I2 R.C/s = I2 R.C/2lmt*h
R = the resistance of coil having ‘T’ number of turns
a = cross section area of wire in mm2 and
ρ =resistivity of conductor in ohm per meter length per mm2 area
R = (ρ* lmt* T)/ a
Now let coil space factor Kc which is the ratio of cross section area of copper
in the coil to total cross section area occupied by coil including insulation
Kc = T * a / Aw
(Aw = total coil section with insulation)
Aw = T*a/Kc mm2
= T*a*10-6/Kc
Also
Aw = h*d m2;
d= depth of the coil in meters
From above values of Aw
a = (h*d*Kc*10-6)/T mm2
so
R = (ρ* lmt* T2)/ (h*d*Kc) ohms
This will give temperature rise θm as