Q2Electrical Machine Design

The output equation of an induction motor relates its power output to its principal dimensions (bore diameter D and core length L) and its specific loadings, and is derived as follows.

Step 1 — EMF equation: the induced EMF per phase in the stator winding is E = 4.44 f Φ Tph Kw, where Φ is flux per pole, Tph is turns per phase, and Kw is the winding factor. The flux per pole is related to the average air-gap flux density Bav and the pole pitch/core length by Φ = Bav × (πD/P) × L, where P is the number of poles.

Step 2 — Specific electric loading: the total ampere-conductors around the periphery is ac × πD, and since there are 2×Tph×m conductors total (for m phases, each conductor carrying current I, counting both go and return conductors per turn), the relationship between ac and the winding current is:

Step 3 — Combine to form the output (kVA) equation: the apparent power input (approximately equal to the kVA rating, ignoring losses for this idealized derivation) for an m-phase machine is Q = m·E·I ×10^-3 (kVA). Substituting the expression for E from Step 1 and I from the ac relation in Step 2, and simplifying (the algebra combines the π, 4.44, and phase-number factors into the numerical output coefficient constant, conventionally 11 for a 3-phase, 50Hz machine when D, L are in meters, Bav in Wb/m², ac in ampere-conductors/m, and n in rev/sec):

where n is the synchronous speed in revolutions per second (n=f/(P/2)×... more precisely n = 2f/P for the rotating field speed in rev/sec), Kw is the stator winding factor, and the numerical constant 11 arises from combining 4.44 (EMF constant), π (pole-pitch-to-diameter relation), and the phase-number/conductor-counting factors for a standard 3-phase machine. This is the standard form of the induction motor output equation used throughout the design process to determine D²L from the specified kVA, speed, and chosen specific loadings, with the split between D and L subsequently determined via an appropriate L/D (or pole-pitch-to-core-length) ratio chosen from empirical design guidelines.

Definition of window space factor (as requested, though primarily a transformer design term): window space factor Kw (sometimes denoted Sf, distinct from the winding factor also denoted Kw above — care must be taken to distinguish the two uses of this symbol across different parts of machine design) is defined, in the context of transformer design, as the ratio of the net conductor copper cross-sectional area within the core window to the total (gross) window area, Kw = (copper area)/(window area), and accounts for the fact that only a fraction of the available window space is actually occupied by current-carrying copper, the remainder being taken up by winding insulation, clearances, cooling ducts, and coil formers, with typical values ranging from about 0.09 for very high-voltage windings (needing thick insulation) to 0.3-0.4 for low-voltage windings.