Q4Electrical Machine Design

This question is effectively identical in content to the transformer main-dimensions and winding-design question appearing in the 2023 (5E1365) paper of this same subject, and the complete step-by-step methodology is as follows.

Steps for determination of main dimensions: (1) compute the required net iron (core) area Ai from the chosen/estimated voltage per turn Et and flux density Bm via Ai=Et/(4.44fBm); (2) apply the area-utilization coefficient appropriate to the chosen core cross-section shape (square: Ai≈0.45d²; 2-stepped cruciform: Ai≈0.62d²; 3-stepped: Ai≈0.75d², where d is the circumscribing circle diameter) to determine d and hence the core step widths; (3) size the yoke cross-section, typically 1.0-1.2 times the limb net iron area (achieved by keeping the yoke flux density somewhat lower than the limb flux density) to control core loss and avoid corner saturation effects; and (4) determine the window height and width from the required winding conductor cross-sectional areas (from load current and current density) plus insulation clearances, cross-checked against the window area implied by the kVA output equation and the assumed window space factor Kw.

Design of low-voltage winding: placed innermost (closest to the core), since it requires the least insulation clearance to the grounded core; typically constructed as cylindrical windings (helical, for very high current/low turns; or multi-layer cylindrical, for more moderate currents), chosen for straightforward construction, good mechanical strength (important since the innermost winding typically experiences the largest radial short-circuit forces, being pushed outward against the supporting structure by electromagnetic forces during a fault), and cost-effectiveness.

Design of high-voltage winding: placed outside the LV winding (separated by an insulating cylinder providing the necessary clearance, which also forms the primary leakage flux path determining the transformer's leakage reactance); typically constructed as disc windings (continuous or interleaved disc type) for larger transformers, providing good mechanical strength against short-circuit forces and improved impulse (lightning/switching surge) voltage distribution along the winding — particularly important given the HV winding's higher voltage stress and correspondingly greater vulnerability to transient overvoltage-induced insulation failure — or as simpler cross-over (coil) windings for smaller HV windings with lower current and correspondingly lower mechanical force requirements.

Elaboration on Step 1 — choosing the voltage per turn: for a first design pass, the voltage per turn is commonly estimated using the empirical relation Et = K√Q, where Q is the rating in kVA and K is an empirical constant depending on transformer type and construction (typically about 0.75-0.85 for 3-phase core-type power transformers, 1.0-1.2 for single-phase core type, and higher for shell type), reflecting the accumulated design experience that larger transformers economically support a higher volt-per-turn (and hence fewer, heavier turns); the value obtained is then refined once the core dimensions are chosen, since Et and Ai are linked through the flux density via Et = 4.44fBmAi.

Elaboration on Step 2 — practical core proportioning: having computed the circumscribing circle diameter d from the net iron area and the chosen stepping arrangement, the designer selects the actual lamination strip widths for each step from standard rolled widths, verifies the achieved net area against the requirement, and fixes the limb pitch (center-to-center leg spacing) as d plus the radial build of both windings plus all clearances — this leg spacing, together with the window height, defines the complete core frame from which the core weight (and hence iron loss, via the material's watts-per-kg loss figure at the design flux density) is directly computed.

Elaboration on Step 4 — window aspect ratio: the window height-to-width ratio is typically chosen in the range 2.5-4 for core-type power transformers; a taller, narrower window shortens the mean turn length (reducing copper weight and I²R loss) but lengthens the core limbs (increasing iron weight and core loss), so the aspect-ratio choice is effectively a copper-versus-iron cost/loss trade-off, often tuned so the design approaches the classical condition for maximum efficiency where iron loss and full-load copper loss are of comparable magnitude.

Conductor selection detail for both windings: once each winding's conductor cross-section is fixed from its current and the chosen current density, the conductor's shape is selected — round wire for small cross-sections (a few mm²), rectangular strip for larger sections (giving better slot/window space utilization), and multiple parallel-connected strips (transposed along the winding to equalize current sharing and suppress circulating eddy currents) once the single-strip size would exceed practical bending or eddy-loss limits, with continuously-transposed cable (CTC) used for the largest LV windings; each conductor is paper-covered (or enamel-covered for small wires) with covering thickness graded to the voltage stress between adjacent turns, which differs substantially between the LV winding (low inter-turn voltage) and HV winding (higher inter-turn and inter-section voltages, especially in the line-end sections that face the steepest impulse-voltage gradients and therefore often receive reinforced insulation).

Closing the design loop: the completed core/window/winding dimension set is finally verified by computing the resulting leakage reactance (from the winding radial builds, gap, and height via the standard leakage-flux formulas), the losses and efficiency, and the temperature rise for the chosen cooling arrangement; any shortfall against the specified impedance, efficiency, or thermal limits sends the design back to adjust Et, Bm, the window aspect ratio, or the current density, exactly as described in the transformer design flowchart discussed elsewhere in this paper — emphasizing that the main-dimension steps and the winding design procedure form one coupled, iterative process rather than two independent calculations. The mark of a sound final design is that every quantity — flux density, current density, window fill, impedance, losses, and temperature rise — lands simultaneously within its specified band, and reaching that state almost always requires two or three passes around this loop rather than a single forward calculation, however carefully the initial design constants were selected — a convergence discipline that applies with equal force to the manual step-by-step design procedure described here and to its automated computer-aided design implementations discussed at length elsewhere in this paper, since both approaches must ultimately satisfy exactly the same underlying coupled physical constraint set governing every practical transformer design, from the smallest distribution unit to the largest generator step-up transformer.