As the end rings are placed in air, although rather close to the motor laminated stack, the skin effect in them is routinely neglected. However, there are applications where the value of slip goes above unity (S = up to 3.0 in standard elevator drives) or the slip frequency is large as in high frequency (high speed) motors to be started at rated frequency (400 Hz in avionics).
For such cases, the multilayer approach may be extended to end rings. To do so we introduce radial and circumpherential layers in the end rings (Figure 9.16) as shown in Reference [7].
Figure 9.16 Bar-end ring transition
a.) slot cross – section b.) radial and circumpherential layer end ring currents
c.) geometry of radial and circumpherential end ring layers
In all layers, the current density is considered uniform. It means that their radial dimension has to be less than the depth of field penetration in aluminum.
The currents in neighboring slots are considered phase-shifted by 2πp1/Nr radians (Nr–number of rotor slots).
The relationship between bar and end ring layer currents (Figure 9.16) is
j2πp1
Ib( j) + Ir(j 1)+ + Ir(j)e Nr = Ir(j) + Ic( j) (9.47)
Figure 9.17 End ring skin effect resistance coefficient KRr
−j2πp1
Rc( j) Ic(j) + R r(j) Ir(j)e Nr = Rc(j 1)− Ic(j 1)− + R r( j) Ir( j) (9.48)
The circumpherential extension of the radial layer rj is assigned a value at start. Now if we add the equations for the bar layer currents, we may solve the system of equations. As long as the radial currents increase, γj is increased in the next iteration cycle until sufficient convergence is met. Some results, after [2], are given in Figure 9.17.
As the slot total height is rather large (above 25 mm), the end ring skin effect is rather large, especially for rotor frequencies above 50(60) Hz. In fact, a notable part of this resistance rise is due to the radial ring currents which tend to distribute the bar currents, gathered toward the slot opening, into most of end ring cross section.