Q10Computer Aided Design of Electrical Machines

(a) Cogging Phenomenon in Squirrel Cage Induction Motor

Cogging (also called magnetic locking, or cogging effect) is a specific starting-condition problem occurring in squirrel-cage induction motors, in which the rotor fails to start rotating (or accelerates only with great difficulty, remaining effectively 'locked') when voltage is first applied, due to strong harmonic magnetic interaction (magnetic locking torque) between the stator and rotor teeth/slots, particularly pronounced when the number of stator slots is equal to (or otherwise unfavorably related to) the number of rotor slots.

Physical mechanism: the air-gap permeance (and hence the local magnetic flux distribution) of a slotted machine varies periodically around the periphery due to the alternating tooth/slot structure on both the stator and rotor surfaces, producing harmonic components in the air-gap flux distribution in addition to the desired fundamental rotating field. When the number of stator slots equals the number of rotor slots (or is related to it by certain other specific unfavorable ratios), these harmonic permeance variations on the stator and rotor sides can align and reinforce each other at specific relative rotor positions, creating a strong reluctance-torque-like magnetic locking effect that tends to hold the rotor teeth aligned directly opposite the stator teeth (the position of minimum magnetic reluctance for these harmonic flux components) — at standstill, if the rotor happens to settle into (or is started from) one of these magnetically-locked alignment positions, the resulting locking torque can be strong enough to prevent the motor from developing sufficient net starting torque to break free and accelerate normally, causing the motor to remain effectively stalled (cogging) despite the applied stator voltage and current.

Why Stator and Rotor Slot Numbers Must Never Be Equal

Because the cogging phenomenon described above arises specifically from the reinforcing alignment of stator and rotor slot-harmonic permeance variations, and this reinforcing alignment condition is most severe and most directly triggered when the stator slot number exactly equals the rotor slot number (since in this case, every stator tooth can simultaneously align with a corresponding rotor tooth at some single rotor angular position, maximizing the locking effect), induction motor designers deliberately and always ensure the number of rotor slots differs from the number of stator slots by an appropriate margin, following established design guidelines that also avoid certain other specific unfavorable slot-number combinations (relating to particular harmonic orders that can similarly cause locking torques, or that can cause other related starting problems such as crawling, an analogous phenomenon caused by certain space harmonics producing an asynchronous torque at a fractional sub-synchronous speed rather than a standstill-locking torque). Avoiding equal (and certain other specifically unfavorable) stator and rotor slot number combinations is therefore one of the most fundamental and universally-followed design rules in induction motor slot-number selection, applied without exception in virtually all squirrel-cage induction motor designs to guarantee reliable self-starting behavior across the motor's full range of possible rotor resting positions at standstill.

Crawling as a Related Starting Phenomenon

Crawling is a distinct but related starting-performance defect, in which the motor, instead of remaining locked at standstill as in cogging, does succeed in starting but accelerates only up to a low sub-synchronous speed (characteristically around one-seventh of synchronous speed) and then 'crawls' at this speed under load, failing to accelerate further up to its normal running speed near synchronous speed. Crawling arises because a non-sinusoidal (distorted) air-gap flux wave, produced by an imperfectly-distributed or imperfectly-pitched stator winding, contains space harmonics (particularly a strong third-space-harmonic, i.e., seventh-order-related, component in common winding arrangements) in addition to the desired fundamental rotating field; each such space harmonic produces its own independent asynchronous induction-motor torque-speed characteristic, with its own synchronous speed equal to Ns/h (h being the harmonic order), and if the resulting harmonic torque's own synchronous speed is low enough and its torque magnitude significant enough, the combined resultant torque curve can develop a stable, saddle-like dip that traps the rotor at that harmonic's sub-synchronous speed under load. Both cogging and crawling are therefore consequences of unwanted harmonic content in the air-gap field (cogging from slot-permeance harmonics, crawling primarily from winding-distribution space harmonics), and both are addressed at the design stage rather than left to be corrected afterward.

Skewing of rotor slots as a mitigation technique: in addition to avoiding unfavorable stator/rotor slot-number combinations (which addresses cogging) and using a well-distributed, short-pitched stator winding (which reduces the space harmonics responsible for crawling), skewing the rotor slots — machining or casting the rotor slots at a small helical angle along the rotor's axial length, typically by about one stator slot pitch over the rotor's length, rather than parallel to the shaft axis — is a further, complementary standard design measure applied to squirrel-cage rotors. Skewing ensures that the rotor bars are never simultaneously aligned with the stator teeth/slots along their entire axial length at any single rotor angular position, since different axial sections of each skewed bar face a continuously varying position relative to the stator slotting; this smooths out the harmonic permeance variation responsible for cogging and simultaneously reduces the harmonic EMFs/torques responsible for crawling, at the modest cost of a slightly increased effective rotor leakage reactance and a small associated reduction in maximum (pull-out) torque. Because skewing addresses both cogging and crawling simultaneously through a single simple manufacturing modification, it is very widely adopted as standard practice in squirrel-cage induction motor design, typically applied in conjunction with, rather than instead of, the slot-number-combination and winding-distribution precautions discussed above.

(b) Design and Working of Stator Core

Construction: the stator core is built from thin (typically 0.35-0.5mm), mutually-insulated silicon-steel laminations, individually punched (or, for large machines, segmented) with the required slot pattern around the inner bore, then stacked axially and clamped tightly together (using end-plates and through-bolts, or welding for smaller cores) to form the complete cylindrical core assembly — the use of thin, insulated laminations (rather than a single solid core) is essential to minimize eddy-current losses induced by the alternating stator flux, since eddy-current loss increases with the square of lamination thickness, making thinner laminations progressively more effective at reducing this loss component at the cost of increased manufacturing complexity and reduced net iron stacking factor.

Slot design: the stator core's inner bore is punched with a number of slots (chosen following the design guidelines discussed in part (a), including avoiding equal stator/rotor slot numbers) to house the distributed three-phase stator winding, with slot shape (commonly semi-closed or open slots, with a narrow slot opening at the bore surface to reduce air-gap flux pulsation/harmonic content while still allowing winding insertion) and slot dimensions chosen based on the required conductor cross-sectional area (determined by the design current density and rated current) and the specified number of slots per pole per phase.

Working (function of stator core): the primary function of the stator core is to provide a low-reluctance path for the magnetic flux established by the stator winding's three-phase alternating currents, which — due to the spatial displacement of the three-phase windings around the periphery (typically 120 electrical degrees apart) combined with the temporal phase displacement of the three-phase currents themselves — combine to produce a rotating magnetic field of constant magnitude, revolving around the stator bore at synchronous speed (Ns=120f/P). This rotating stator field sweeps past the rotor conductors (or rotor bars, for a squirrel-cage rotor), inducing EMFs and currents in them (by electromagnetic induction, giving the induction motor its name), and the interaction between the rotating stator field and the induced rotor currents produces the torque that drives the rotor to rotate at a speed slightly below synchronous speed (the slip speed difference being essential to sustain the induced rotor EMF and current, and hence the torque-producing interaction, in a standard induction motor). The stator core's low-reluctance magnetic path (compared to air) is essential to this entire process, since it allows the required rotating flux to be established with a comparatively modest magnetizing current, directly analogous to the role of the core in any other transformer or rotating-machine magnetic circuit, while the core's laminated construction (as discussed above) minimizes the core (iron) losses that would otherwise be incurred as this alternating/rotating flux continuously varies within the core material during normal motor operation.