For small or high transverse resistivity (Rq) expressions (11.87 – 11.90), use the computation of transverse and cage no-load stray losses with Emν from
(11.78) and ν = Ns ± p1. It has been shown that for this case, with ν ≈ Ns, the transverse losses have a minimum of 0.9 < Nr/Ns < 1.1.
It means that for low contact (transverse) resistivity, the rotor and stator slot numbers should not be too far away from each other. Especially for Nr/Ns < 1, small transverse rotor losses are expected.
11.8. LOAD ROTOR SKEWED NONINSULATED CAGE LOSSES
While under no-load, the airgap permeance first harmonic was important as it acted upon the airgap flux distribution; under load, it is the stator mmf harmonics of same order Ns ± p1 that are important as the stator current increases with load. This is true for chorded pitch stator windings when the 5th and 7th pole pair phase belt harmonics are small.
The airgap flux density Bν will now be coming from a different source:
Bν = pν1 KKww1ν gK 1µc (0F+1pKs ) (11.92)
F1p is the stator mmf fundamental amplitude.
For full pitch windings however, the 5th and 7th (phase belt) harmonics are to be considered and the conditions of low or high transverse resistivity (11.85) has to be verified for them as well with ν = 5p1, 7p1.
Transverse cage losses for both smaller or higher transverse resistivity Rq are inversely proportional to differential leakage coefficient τdν which, for ν =
Ns, is
πNs 2 N
τdNs = πrNs 2 −1 (11.93)
sin2 Nr
It may be shown that τdNs increases when Ns/Nr > 1.
Building IMs with Ns/Nr > 1 seems very good to reduce the transverse cage losses with skewed noninsulated rotor bars. For such designs, it may be adequate to even use non-skewed rotor slots when the additional transverse losses are almost zero.
Care must be exercised to check if the parasitic torques are small enough to secure safe starts. We should also notice that skewing leads to a small attenuation of tooth flux pulsation core losses by the rotor cage currents.
In general, the full load transverse cage loss Pdn is related to its value under no-load Pd0 by the load multiplication factor Cloads (11.53), for chorded pitch stator windings.
Pdn = P Cd0 loads (11.94)
For a full pitch (single layer) winding Cloads has to be changed to add the 5th and 7th phase belt harmonics in a similar way as in (11.60 and 11.61).
Kw5 2 (1+ τdN )
2
Pdn = P Cd0 loads 1+ ξ(2NKs −w1P n1 )+ ξ(1(2N+ τs +d6P n1s)) (11.95)
with ξ(Ns−p1)n and ξ(Ns+p1)n from (11.51).
Example 11.5. For the motor with the data in Example 11.4, let us determine the Pdn/Pd0 (load to no-load transverse rotor losses).
Solution
We are to use (11.95).
From Example 11.4, ξ(Ns−p1)n = 3.6/ Kc , Kc = 1.8, Kw5 = 0.2, Kw1 = 0.965, Cloads = 1.0873, ξ(Ns+p1)n = 2.8/ Kc , 1+ τdNs = 43 , 1+ τd6 =1.37 .
We have now all data to calculate
2 0.2 2 43
2 0.2 2 43
PPd0dn =1.08731+ 3.6 0.9652 + 2.81.372 =1.54! 1.8 1.8