Q1Electrical Machine Design

Given data: P=90kW, V=500V (line), f=50Hz, 3-phase, 8-pole, star-connected stator, 63 stator slots, 6 conductors/slot, target rotor open-circuit slip-ring (line) voltage = 400V, motor efficiency η=0.9, power factor=0.86.

(a) & (b) Number of rotor slots and conductors per slot: first compute the stator turns/phase: total stator conductors = 63×6=378, conductors/phase=378/3=126, turns/phase Tstator=126/2=63. Stator phase voltage (star) = 500/√3 ≈ 288.7V. Target rotor phase voltage (star) = 400/√3 ≈ 230.9V. Turns ratio = 288.7/230.9 ≈ 1.25, giving required rotor turns/phase = 63/1.25 = 50.4 ≈ 50 (rounded to a practical integer). Total rotor conductors = 50×2×3 = 300. Choosing the rotor slot number to avoid cogging (S2≠S1=63) and unfavorable crawling ratios (S2/P should not be a small integer; with P=8, avoid S2=8,16,24,... and other flagged combinations) — a practical choice close to the stator slot count while satisfying these constraints is S2=54 slots, giving conductors per slot = 300/54 ≈ 5.6, rounded to a practical even integer of 6 conductors/slot (adjusting the exact turns count slightly from the idealized 50 to the practically realizable value consistent with an integer number of slots and conductors/slot, a normal, expected step in real winding design, since fractional conductor counts are not physically realizable).

(c) Coil span: with 8 poles and 54 rotor slots, slots per pole = 54/8 = 6.75; choosing a full-pitch (or near-full-pitch) coil span of approximately 7 slots (rounded to the nearest practical integer close to the slots-per-pole value) is a typical choice for a wound rotor winding of this type.

(d) Slip-ring voltage on open circuit: by design target, this is 400V (line), as specified — recomputing from the actual, rounded rotor turns count (T=50, giving conductors/slot=6 and total conductors=54×6=324, turns/phase=324/2/3=54) would give a turns ratio of 63/54≈1.167, and hence an actual achieved open-circuit rotor line voltage of ≈500×(54/63)≈428V — illustrating that the practical rounding of slots/conductors to integer, standard values causes the achieved voltage to deviate slightly from the exact 400V target, requiring the designer to either accept this small deviation or iterate the slot/turns selection further to converge more closely on the exact target voltage.

(e) Approximate full-load current per phase in rotor: stator full-load line current from the motor's power/efficiency/power-factor:

Referring this to the rotor side approximately via the turns ratio (rotor current ≈ stator current × turns ratio, for a wound rotor at or near standstill/low-slip reference):

giving an approximate full-load rotor current per phase in the range of roughly 155-170A depending on the precise final rounded turns ratio adopted, which together with the chosen rotor current density then determines the required rotor bar/conductor cross-sectional area, completing this rotor winding design estimate.

Why the rotor slot count must differ from 63: it is worth noting explicitly why the rotor cannot simply reuse the stator's 63-slot count: equal stator and rotor slot numbers (S1=S2) produce cogging (magnetic locking at standstill), as discussed elsewhere in this paper, making S2=63 categorically unacceptable. Additionally, with P=8 poles, slot-combination rules require avoiding S1−S2 = 0, ±P, ±2P, ±5P (i.e., differences of 0, ±8, ±16, ±40) to prevent cogging, synchronous crawling, and noisy operation — the chosen S2=54 gives S1−S2=9, which avoids all of these flagged differences, illustrating how the slot-combination selection rules are applied in a concrete design decision rather than merely stated abstractly.

Verification of the winding's electrical balance: with S2=54 slots, 8 poles, and 3 phases, the rotor winding has q = 54/(8×3) = 2.25 slots per pole per phase — a fractional-slot winding, which is acceptable for a wound rotor provided the winding layout is designed to keep the three phases balanced (identical total turns and matched phase-belt distributions per phase); fractional-slot windings are routinely used in practice and can even offer benefits in suppressing certain space harmonics, though they require more careful winding layout design than simple integral-slot windings, which is part of the detailed winding design work that follows the main slot/turns selection performed here.

Thermal sizing implication of the computed rotor current: the approximate 155-170A full-load rotor phase current, taken with a typical wound-rotor conductor current density of around 3-5 A/mm², implies a required rotor conductor cross-section of roughly 35-55 mm², which must physically fit (together with slot insulation appropriate to the ~430V rotor voltage class) within the chosen 54 slots — a final geometric consistency check tying together the electrical (turns/voltage), magnetic (slot/tooth flux density), and thermal (current density) aspects of the rotor design, and completing the full set of quantities the question requests: (a) 54 rotor slots, (b) 6 conductors per slot, (c) coil span ≈ 7 slots, (d) open-circuit slip-ring voltage ≈ 400-430V depending on final rounding, and (e) approximately 155-170A full-load rotor current per phase.

General principle this problem illustrates: wound-rotor winding design is fundamentally a constrained rounding exercise — the exact turns count demanded by the target slip-ring voltage almost never coincides with an integer number of conductors per slot in a slot count that simultaneously satisfies the anti-cogging/anti-crawling combination rules, so the designer must select the nearest feasible integer combination and then verify that the resulting deviation in achieved slip-ring voltage (here, roughly 400V versus 428V depending on the rounding direction chosen) remains within the tolerance the external rotor-circuit equipment can accept. When it does not, the next adjustment lever is the stator winding itself (revisiting the stator conductors per slot or slot count), showing that stator and rotor winding designs, while performed sequentially, are ultimately coupled through the voltage-ratio requirement and may both need revision in a final converged design. The wound-rotor construction assumed throughout this problem is itself a deliberate specification choice for a 90kW, 8-pole machine of this class, made precisely so that external rotor resistance can be used to deliver high starting torque at moderate starting current — the specific practical application context that ultimately gives the slip-ring voltage and rotor current calculations performed throughout this entire problem their real practical engineering purpose and design significance.