Hysteresis and eddy current losses; loop area

Hysteresis loss. When the flux in a magnetic material alternates, its magnetic domains are reversed again and again. They resist this because of internal friction, so the flux density B lags behind the field strength H. The energy spent in each cycle of reversal appears as heat in the core. This is the hysteresis loss.

Steinmetz's formula: Pₕ = ηₕ · B_max^1.6 · f · V (watts), where ηₕ is the Steinmetz coefficient of the material, f the frequency and V the volume of the core.

Eddy current loss. An alternating flux induces emf in the core itself. Since the core is conducting, this emf drives circulating currents (eddy currents) in planes perpendicular to the flux. They cause I²R loss and heat the core.

Pₑ = kₑ · B_max² · f² · t² · V (watts), where t is the thickness of the lamination and kₑ a constant depending on the resistivity of the material.

Proof: hysteresis loss = area of the B–H loop

Take a core of length l, area a and N turns, carrying current i, with field H = Ni/l.

emf induced         e = N dφ/dt = N a dB/dt
energy in time dt   dW = e i dt
                       = N a (dB/dt) · (H l/N) · dt
                       = (a l) H dB

Here a·l = V is the volume of the core, so the energy per unit volume is dW/V = H dB.

For one complete cycle the total energy per unit volume is

W = ∮ H dB = area enclosed by the B–H loop (J/m³ per cycle)

At f cycles per second, Pₕ = f · V · (loop area) watts. A larger loop means a larger loss; this is why soft magnetic materials with a narrow loop are used for cores.