Design and Voltage Supply of High-Speed Induction - Aaltodoc

l = + , (7.2)
sλs 2leλ e l wλ
w
where le and lw are the lengths of a coil end and a circular part of the end winding. λe and λw are
empirical coefficients depending on the type of the winding system.
The end ring resistance and inductance of the squirrel cage are considered in a similar manner. In
the case of a coated rotor, the coating is divided into bars and end ring parts and the same model can
be used. The resistance of an end ring is calculated based on its dimensions and conductivity. In the
initial FEM model, the inductance of the end ring was calculated using empirical equations for a
thick cylindrical coil in air with a constant current density in the coil. These equations were
presented by Foelsch (1936) with an error of less than 0.3 %:
er
0
2
c
L = N a λ , (7.3)
c
f
where Nc is the number of turns and ac is the average diameter of the coil. For the end ring, Nc
equals one. λf is a coefficient depending on the dimensions of the coil. A total of eight equations for
λf are needed to cover all the dimensions with the accuracy mentioned above.
For the high-speed rotors considered, this however does not give accurate results. The end rings are
mounted on the surface of a solid ferromagnetic iron. Thus, the real values for the end ring
inductance are higher than those given by Eq. 7.3. Also, the Foelsch method supposes that the
current is constant in amplitude and direction around the coil. In reality, there is something close to
a sinusoidal current coverage around the end ring.
To get another approximation for the end ring inductance, a separate 2D FEM calculation was made
using free 2D FEM software called FEMM (Meeker 1999). In this case, a differential piece of an
end ring was modeled in a planar coordinates system. Fig. 7.1 shows a quarter cross-section of the
motor with a copper coated solid steel rotor. The magnetic field is induced by the end region
currents in the stator and the rotor. The bottom boundary has the Neumann boundary condition and
the right boundary has the Dirichlet boundary condition set. The rotor is situated at the bottom. The
copper coating lies at the surface of the rotor. The end ring part of the coating is darkened. The
other half of the stator stack, divided by the cooling duct, is on the right and the stator end winding
is next to it in the center. On the left, there is a solid iron stator of the axial magnetic bearing.
Considering the small slip and the lamination of the stator core, the conductivities of the stator and
rotor parts were zeroed. Only in the stator of the axial bearing can some eddy currents be seen
flowing. No skin effect for the end ring or saturation of the iron parts was modeled. In the
calculation for the inductance, only the end ring current was used to determine the self-inductance.
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