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# Three phase transformer losses

Unbalanced load is connected to the three phase transformer.

Problem type:
Plane problem of AC magnetics.

Geometry:

Given:
Core permeability μ = 1000,
Core mass density ρ = 7650 kg/m3,
Core losses Cm = 1.5 W/kg (at f=50 Hz and B=1.5 T),
Copper conductivity g=56e6 S/m,
Primary winding Y: 6.25 mm2 x 2560 turns,
Secondary winding Δ: 16 mm2 x 150 turns,
Frequency f=50 Hz.

Calculate magnetic and electric losses in the three phase transformer.

Solution:
Winding (copper) losses volume density:
pe = j2 / g [W/m3].
Steinmetz equation to calculate core (steel) losses volume density:
pm = Cm · (f/50)α · (B/1.5)β · ρ [W/m3],
where α = 1, β = 2, B - average flux density in the core (peak value).

Results:
Average flux density in the core (peak value): B = √0.746 · √2 = 1.22 T.
Power losses in the core pm = 1.5 · (50/50)1 · (1.22/1.5)2 · 7650 = 7.59 kW/m3.

Winding (copper) losses volume density
pe = (I/S)2 / g:

 Windinng name Conductor cross section, S Phase current (RMS), I Joule heat losses, pe A1 6.25 mm2 19 A 165 kW/m3 B1 6.25 mm2 13.4 A 82 kW/m3 C1 6.25 mm2 9.4 A 40 kW/m3 A2 16 mm2 45.7 A 146 kW/m3 B2 16 mm2 35.3 A 87 kW/m3 C2 16 mm2 23.2 A 38 kW/m3

* Reference: http://en.wikipedia.org/wiki/Transformer