Q3Electrical Machine Design

This problem illustrates the phenomenon of synchronous crawling (also called synchronous hooking) caused by an unfavorable combination of stator and rotor slot numbers, a well-known design pitfall that the slot-number selection rules in induction motor design are specifically intended to avoid.

Step 1 — Synchronous speed of the fundamental field:

Step 2 — Origin of tooth-ripple (slot) harmonics: in addition to the desired fundamental rotating field, the discrete distribution of stator and rotor windings into a finite number of slots produces space harmonics in the air-gap MMF/flux wave, known as tooth-ripple or slot harmonics. The order of the most significant slot harmonics produced by the rotor slotting is given by:

where S2 is the number of rotor slots and P is the number of poles. For this motor, S2=28 and P=4, giving S2/P = 7. Taking k=1, the relevant harmonic orders are h = 7-1 = 6 and h = 7+1 = 8; however, the specific harmonic responsible for producing a rotating field that can synchronously interact with the rotor and cause crawling at a sub-multiple of synchronous speed is directly related to the ratio S2/P = 7 itself (this ratio being an integer is precisely the unfavorable condition that creates the risk of synchronous crawling, since it means the rotor slotting produces a spatial harmonic pattern that repeats synchronously with the fundamental field pattern after accounting for the pole-pair structure).

Step 3 — Speed at which the harmonic field is stationary relative to the rotor: a harmonic field of order h rotates at a speed of Ns/h relative to the fundamental synchronous speed reference frame. For h=7 (arising from S2/P=7):

This exactly matches the speed given in the question, confirming that the 7th-order space harmonic field (arising because S2/P = 28/4 = 7 is an integer) can produce a synchronous locking torque at N = 214.3 rpm — if the motor's normal induction-motor torque is weak at this low speed (as is often the case near start-up, where slip is very high and starting torque may be relatively low), the parasitic synchronous torque from this harmonic field can dominate and cause the rotor to 'lock' or 'crawl' at this low sub-synchronous speed instead of accelerating up to its normal running speed near Ns, a serious motor starting fault.

Design implication: this problem demonstrates precisely why induction motor design practice includes strict rules for selecting the rotor slot number S2 relative to the stator slot number S1 and the pole number P — specifically avoiding S2/P (and related combinations such as S1-S2, S1+S2 relative to P) being equal to small integers like 1, 6, 7, or other values known to produce strong synchronous-crawling or cogging harmonics, which is why in practice a designer would revise this rotor slot number (e.g., to 22 or 34 rather than 28) to avoid this specific 7th-harmonic crawling condition identified by the calculation above.