Q6Computer Aided Design of Electrical Machines

(a) Power Transformer vs Distribution Transformer

Power transformers are used at generating stations and major bulk-power transmission/sub-transmission substations, to step voltage up (at the generating end) or down (at major receiving substations) for efficient long-distance power transmission, and are characterized by very high individual ratings (commonly tens to hundreds of MVA), continuous near-full-load operation with a high load factor (since they primarily serve bulk transmission capacity that is utilized close to its designed rating for most of its operating time), and design optimization primarily aimed at minimizing full-load copper (I^2R) losses, since these losses dominate total losses when the transformer operates continuously at or near its rated load.

Distribution transformers, by contrast, are used at the final stage of the power distribution network, stepping down from a medium sub-transmission/distribution voltage to the final low utilization voltage supplied directly to end consumers (residential, commercial, and small industrial loads), and are characterized by comparatively lower individual ratings (typically a few kVA to a few hundred kVA, as in the design problem in this paper), a much lower and highly variable load factor (since actual instantaneous consumer load fluctuates considerably throughout the day, often averaging well below the transformer's rated capacity), and near-continuous energization (remaining connected to the supply, and hence continuously incurring no-load/core losses, for essentially 24 hours a day regardless of whether any actual load current is flowing at a given moment).

Design Considerations Arising from This Difference

Because a distribution transformer remains continuously energized (and hence continuously incurring core/iron losses) for its entire operating life, but carries significant load current (and hence copper losses) for only a fraction of that time (given its typically low average load factor), distribution transformer design deliberately prioritizes minimizing no-load (core) losses over minimizing full-load copper losses — this is typically achieved by designing with a somewhat lower operating flux density than would be chosen for a power transformer of similar rating (reducing core/hysteresis and eddy-current losses, at some cost in increased core material required for a given voltage-per-turn), and by using higher-grade, lower-loss core lamination material even at correspondingly higher material cost, since the resulting core-loss reduction, accumulated continuously over 24 hours a day for the transformer's entire service life, typically far outweighs the increased material cost — this design philosophy is formally captured in the concept of 'all-day efficiency' (energy-based efficiency calculated over a full day's actual, variable load cycle, rather than simple full-load power efficiency), which is the standard performance metric used specifically for optimizing distribution transformer design, in contrast to the simple full-load efficiency metric more directly relevant to continuously fully-loaded power transformers. Power transformers, by contrast, since they operate at high load factor much of the time, are designed with less concern for minimizing no-load losses (since these represent a smaller fraction of their total accumulated loss over time compared to load-dependent copper losses), and design attention instead focuses more heavily on minimizing full-load copper losses and ensuring adequate cooling capacity to handle their much larger absolute loss magnitudes at their very high power ratings.

All-Day Efficiency: Illustrative Calculation Approach

All-day efficiency is defined as the ratio of total energy output (in kWh) to total energy input (in kWh) over a full 24-hour period, rather than the ratio of instantaneous power output to power input at a single loading condition used for ordinary (power) efficiency. For a distribution transformer with a known core (iron) loss Pi (constant, since it is energized continuously irrespective of load) and a known full-load copper loss Pcu(FL), operating over a typical daily load cycle broken into intervals of known load fraction x (as a fraction of full load) and known duration in hours, the all-day efficiency is calculated as: eta(all-day)=(sum of output energy over all intervals)/(sum of output energy + core-loss energy over 24 hours + sum of copper-loss energy over all intervals), where the copper loss in each interval is taken as x^2Pcu(FL) (since copper loss varies with the square of the load current) and the core loss Pi24 hours is included regardless of load level. Since the transformer's core loss is incurred for the full 24 hours regardless of the actual load carried, while its copper loss is incurred only during, and in proportion to the square of, the actual load current drawn, minimizing core loss (as prioritized in distribution transformer design) has a proportionally much larger effect on all-day efficiency for a lightly and intermittently loaded distribution transformer than an equivalent reduction in full-load copper loss would, directly justifying the lower flux density and higher-grade core material design choice discussed above for this class of transformer, and explaining why all-day efficiency, rather than simple full-load efficiency, is the correct and standard criterion used to evaluate and optimize distribution transformer designs of the kind illustrated numerically in part (b) of this question.

Core-Type vs Shell-Type Construction

The two fundamental transformer core constructions differ in how the core and windings are geometrically arranged relative to each other. In core-type construction (as assumed for this 200kVA design), the windings (low-voltage winding placed nearer the core, with the high-voltage winding wound concentrically over it) surround the core limbs, with the magnetic circuit forming a simple rectangular (or, for three-phase units, three-limb) loop passing through the core limbs and yokes; core-type construction is mechanically simpler, generally preferred for higher-voltage applications (since the concentric winding arrangement more easily accommodates the greater inter-winding insulation clearance required), and is the more common construction for medium and large power and distribution transformers. In shell-type construction, by contrast, the core surrounds the windings (the windings are sandwiched between core limbs, with the core split into laminated sections that enclose the winding on more than one side), providing a mechanically more rigid structure with a naturally shorter, lower-reluctance magnetic path and improved short-circuit-force withstand capability, but with a comparatively more complex core-and-coil assembly process; shell-type construction is more commonly favored for lower-voltage, larger-current applications and for some specialized high-current or high-mechanical-stress applications where its superior bracing against winding forces is particularly valuable.

(b) Numerical: Overall Core Dimensions of 200kVA 3-Phase Transformer

Given: Q=200 kVA, f=50 Hz, 3-phase, delta-star connected, maximum flux density Bm=1.3 Wb/m^2, current density delta=2.5 A/mm^2=2.5x10^6 A/m^2, window space factor Kw=0.3, overall height=overall width, window area Aw=1.25 times core area Ai.

Step 1: Apply the Three-Phase Transformer Output Equation

For a three-phase core-type transformer, the standard output equation (using the constant 3.33, reflecting the combined per-phase contribution across all three limbs) is:

Substituting the given window-to-core area ratio Aw=1.25*Ai:

Step 2: Core Dimension

Treating the net core cross-section as square (consistent with the design approach used in the companion single-phase transformer problem in this paper), the side of the square core is:

Window area: Aw=1.25Ai=1.250.03139=0.03924 m^2=392.4 cm^2.

Result: the net core cross-sectional area is approximately 313.9 cm^2, giving a core dimension (square core side) of approximately 177.2mm, with a corresponding window area of approximately 392.4 cm^2. The additional condition stated in the problem, that the transformer's overall height equals its overall width, is a further constraint used at the next stage of the complete mechanical/frame design (relating the core's limb spacing, yoke dimensions, and window height/width to arrive at final overall physical dimensions for the tank and frame), building upon this core cross-sectional area result as the foundational input to that subsequent overall-dimensioning step of the full transformer design procedure.