Q4Electrical Machine Design

Derivation of rotor bar current and cross-section: in a squirrel-cage rotor, each bar can be treated as a single-turn 'winding', and by the MMF balance principle (the rotor MMF must balance the stator MMF, similar to the primary-secondary MMF balance in a transformer), the rotor bar current is related to the stator current and the effective number of stator conductors per pole. The standard approximate relation used is:

(where m1, Tph1, Kw1, I1 are the stator phase number, turns per phase, winding factor, and phase current respectively, and S2 is the number of rotor slots/bars), derived from equating the total rotor MMF (Ibar × S2, summed appropriately around the periphery) to the stator MMF. Once Ibar is estimated, the bar cross-sectional area is obtained by dividing by the chosen rotor bar current density (typically higher than the stator current density, since the rotor bars are usually bare, uninsulated conductors with good direct thermal contact to the surrounding iron, allowing more aggressive current loading, commonly 4-7 A/mm² for cage rotors).

Derivation of end-ring cross-section: the end rings, which connect all the rotor bars together at each end of the rotor to complete the cage circuit, must carry the vector sum of the currents from all the bars converging into the ring segment. Using the relation between bar current and the equivalent end-ring current (derived by treating the rotor bars and end-ring segments as forming a polygon of currents, analogous to line-phase current relationships in polyphase systems), the standard approximate design relation is:

where P is the number of poles. This arises because the end-ring effectively carries a current that is the phasor sum of the currents from S2/P bars per pole, spread around 180 electrical degrees, giving the π factor in the denominator via the standard sum-of-sinusoidal-phasors relation. The end-ring cross-sectional area is then obtained by dividing this ring current by the chosen end-ring current density (typically somewhat lower than the bar current density, since the end ring often runs hotter due to receiving heat conducted from all the bars converging into it).

Limitations of squirrel-cage rotor: the squirrel-cage rotor's resistance is fixed by its construction (bar and end-ring material and dimensions) and cannot be varied during operation, unlike a wound rotor where external resistance can be inserted via slip rings. This means a cage rotor design must accept a fundamental trade-off: low rotor resistance gives good running efficiency and low slip at rated load, but produces relatively low starting torque and high starting current (since starting torque is roughly proportional to rotor resistance in the useful low-resistance range, while starting current is inversely related), whereas a design with higher rotor resistance improves starting torque and reduces starting current but at the cost of poorer running efficiency and higher full-load slip. Standard cage designs (NEMA/IS design classes A, B, C, D) represent different fixed points along this trade-off curve, and once a design class is manufactured, its starting/running trade-off cannot be adjusted in the field, unlike a wound-rotor motor's externally adjustable starting resistance — this inflexibility is the fundamental limitation of squirrel-cage rotor construction, compensated in practice by more sophisticated cage designs such as double-cage or deep-bar rotors, which exploit the skin effect to give an effectively higher resistance at starting (high slip frequency) and lower resistance at running speed (low slip frequency), partially overcoming this basic single-cage limitation without requiring external rotor connections.

Deep-bar and double-cage rotor design as a partial solution: the deep-bar rotor uses tall, narrow bars in which the skin effect (at the relatively high rotor-current frequency present at standstill/high slip) forces current to concentrate near the top of the bar, effectively increasing the bar's AC resistance at starting compared to its DC resistance at low-slip running conditions, giving an automatic, frequency-dependent resistance variation without any moving parts or external connections. The double-cage rotor takes this concept further by using two physically separate cages: an outer cage of high-resistance, low-reactance material (positioned near the air gap, dominating current flow at high slip/starting due to its lower reactance path at high rotor frequency) and an inner cage of low-resistance, high-reactance material (positioned deeper in the rotor iron, dominating current flow at low slip/running conditions once the skin effect diminishes at low rotor frequency), giving an even more pronounced and controllable variation of effective rotor resistance with slip than the simple deep-bar design alone.

Further limitations beyond fixed resistance: in addition to the fixed-resistance limitation already discussed, squirrel-cage rotors offer no direct means of speed control via rotor circuit manipulation (unlike wound-rotor motors, where external rotor resistance can also be used for speed control in addition to starting torque improvement, in applications tolerating the associated efficiency loss at reduced speed), meaning cage-rotor induction motors intended for variable-speed applications must instead rely on stator-side control methods such as variable-frequency drives; and the rotor bars/end-rings, being a permanently cast or fabricated structure, cannot be inspected, repaired, or individually replaced in the field in the way a wound rotor's windings can be, meaning a cage-rotor failure (e.g., broken bars from thermal cycling fatigue) typically requires complete rotor replacement rather than localized repair, an important maintenance and lifecycle-cost consideration weighed against the cage rotor's otherwise superior simplicity, ruggedness, and lower manufacturing cost compared to a wound-rotor design.

End-ring thermal and mechanical sizing note: beyond the electrical cross-section derived above, the end-ring design must also verify mechanical integrity under the centrifugal loading of the ring's own mass at maximum overspeed, and thermal capacity during prolonged starting or stalled conditions, when the entire slip-frequency rotor loss is dissipated in the bars and rings with little cooling airflow — these transient-duty checks, rather than the steady-state full-load current alone, frequently govern the final ring cross-section in practice, particularly for motors specified for arduous starting duties such as high-inertia fan or crusher loads, where repeated prolonged starts subject the cage bars and rings to severe, repeated thermal excursions extending well beyond anything ever experienced during ordinary steady full-load running conditions.