Magnetic Circuits of AC Machines: Analysis and Derivations

Assignment, T2, 2019 ELEC 3105
ELEC 3105: Electrical Energy
ASSIGNMENT: T2, 2019
Solution to Question 1
Following inputs are provided,
Air gap length lg =2.5 mm
Number of Poles P = 2
Cross sectional area of the air gap Ag=500 c m2
Effective diameter of the rotor Dr =20 cm
Axial length h=60 cm
Mean length of stator flux path per pole lc ,stator =100 cm
Cross-sectional area seen by the flux in the core ¿ 450 c m2
Effective number of turns of the field coils Nf =500
Section (a)
Electrical machines are made of ferromagnetic materials. Ferromagnetic materials have nonlinear
material characteristics. Hence it is crucial to design the synchronous machine with appropriate field
current. The magnetic characteristic of the ferromagnetic material is represented in figure 1, which is a
plot of flux density versus field intensity.
The B-H curve has three regions, the first region can be categorized as linear region as the flux density
increases linearly with the flux intensity. The last part of the curve can be categorized as saturation region
as even large increase in the magnetic field does not cause any considerable increase in flux density. This
tells us that the relative permeability of the ferromagnetic material is not linear and changes with the
excitation current. The middle portion where the curve starts to bend can be described as knee point of the
curve.
The machine with great flux density can be designed by selecting the operating point very close to Knee
Point of the magnetizing curve.
The machines are operated at knee point considering the fact that at this point flux density of the
ferromagnetic material is more, thus resulting in lesser cross section of the material for construction of
machine.
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Assignment, T2, 2019 ELEC 3105
Figure 1 Magnetizing curve of given ferromagnetic material
Now to select the operating point close to knee of the curve, we can plot the load line. The load line can
be plotted by the equation obtained from the Amperes Circuital law,
i.e.
N f ∗I f
P∗Cf
=Hc , s∗lc ,s + H g∗lg + Hc ,r∗lc ,r + H g∗lg
N f ∗I f
P∗Cf
=Hc , s∗lc ,s + 2∗Bg
μ0
∗lg + Hc , r∗lc ,r
Where,
H g=Field intensity ∈air gap
Hc , s=Field intensity ∈stator
Hc ,r =Field intensity ∈rotor
Cf =No. of ∥circuits
It is given that, the leakage in airgap is negligible, and flux density of stator and rotor are same, also the
stator and rotor are made of same ferromagnetic material.
Hence, the flux density of core and air will be same and magnetic intensity of stator and rotor will be
same
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Assignment, T2, 2019 ELEC 3105
N f ∗I f
P∗Cf
=Hc∗( lc, s +lc ,r ) + 2∗Bc
μ0
∗lg
Bc= μ0
2lg [ Nf ∗I f
P∗Cf
−Hc∗( lc, s +lc ,r ) ]
To design the machine with great flux density, Bis directly proportional to the field current, I f . Hence, the
field current is selected as 22 A DC. By substituting the values in the above equation, the load line from
this equation and the BH curve from the figure 1 is plotted in the same graph and it is observed that
intersects at a point as seen in the figure 2.
Figure 2 Plot to determine the operating point of ac machine
Conclusion:
The co-ordinates at which the two curve intercepts can is indicated in the figure 2 which can be chosen as
the operating point of the ac machine. The X axis indicates field intensity and Y axis indicates.
Hence, the operating value of flux density, B = 1.276 T & and field intensity, H = 353 A/m.
Section (b)
The relation between the magnetic flux density B, and field intensity H is given by the formula
B=μ0 μr H
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Assignment, T2, 2019 ELEC 3105
For the operating point selected in section (a),
B = 1.276 T; H = 353 A/m; μ0 =4 π∗10−7
Hence relative permeability can be calculated as
μr = 1.276
353∗4 π∗10−7 =2876.5
Section (c)
The magnetic circuit can be analyzed from the flux path of the ac machine.
Figure 3 Flux path of ac machine
The dotted line in the above figure indicates the flux path of the ac machine. The magnetic circuit can be
represented in analogy with the electrical circuit representing the reluctances of various parts as below,
Figure 4 Analogy of Magnetic circuit with electrical circuit indicating reluctances
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