Bertotti-Steinmetz Magnetic Core Hysteresis and Eddy Loss Formula

See why classical eddy-current loss overtakes hysteresis as switching frequency rises

Bertotti’s (1988) physical core-loss decomposition splits volumetric magnetic power dissipation P_v into three frequency-scaling mechanisms: static loop Hysteresis P_hyst = k_h * f * B_m^beta (scaling as f^1), lamination Classical Eddy Currents P_eddy = k_e * (f * B_m)^2 (scaling as f^2!), and Anomalous Domain-Wall Excess Loss P_exc = k_exc * (f * B_m)^1.5 (scaling as f^1.5).

Verify on an inductor core at f = 16 kHz (B2) and peak flux density B_m = 0.25 T (B3, so f * B_m = 4.00) with Steinmetz exponent beta = 2.0 (B4), k_h = 30 (B5), k_e = 2.5 (B6), and k_exc = 1.5 (B7): P_hyst = 30 * 16 * 0.0625 = 30.00 kW/m^3, P_eddy = 2.5 * 16 = 40.00 kW/m^3, and P_exc = 1.5 * 8 = 12.00 kW/m^3 — totaling 82.00 kW/m^3! Quadruple frequency to 64 kHz and hysteresis quadruples (120), excess grows 8x (96), while eddy loss explodes 16-fold to 640, pushing total core loss to 856.00 kW/m^3!

Total Three-Term Bertotti Volumetric Core Loss Density Formula

Validate non-negative loss coefficients, positive Steinmetz exponent beta >= 1, and non-negative frequency and flux amplitude.

=LET(f_khz, B2, b_pk, B3, bet_s, B4, k_hy, B5, k_ed, B6, k_ex, B7, IF(OR(f_khz < 0, b_pk < 0, bet_s < 1, k_hy < 0, k_ed < 0, k_ex < 0), "Invalid Core-Loss Input", IF(OR(f_khz = 0, b_pk = 0), 0, LET(p_hy, k_hy * f_khz * POWER(b_pk, bet_s), p_ed, k_ed * ((f_khz * b_pk)^2), p_ex, k_ex * POWER(f_khz * b_pk, 1.5), ROUND(p_hy + p_ed + p_ex, 4)))))

Eddy-Current Fraction of Total Core Loss Formula

Track P_eddy / P_total to decide when thinner laminations, powdered iron, or ferrite materials are required.

=IF(OR(ISTEXT(E4), E4 = 0), 0, ROUND((B6 * ((B2 * B3)^2)) / E4, 4))

Further reading

• Arte Sheets Lesson — RF Skin Depth AC Conductor Penetration Formula
• Microsoft Support — POWER function