Power Magnetics, High-Frequency Planar Transformers & Eddy-Current Proximity Loss
Dowell Transformer AC Winding Resistance (Fr) & Proximity Loss
Calculate High-Frequency Transformer Normalized Foil Thickness Δ = h / δ, Dowell Sub-Skin AC-to-DC Resistance Ratio Fr = 1 + ((5*m² - 1)/45)*Δ⁴, DC Copper Loss (W), and Total AC Winding Loss Before and After P-S-P-S Layer Interleaving.
Debunk the "thinner than 1 skin depth" myth—m² proximity fields dominate multi-layer windings!
Power-electronics engineers designing 50–500 kHz planar and foil SMPS transformers often pick copper foil thickness h smaller than skin depth δ (Δ = h / δ < 1) assuming AC resistance equals DC resistance! Dowell’s classic magnetics analysis proves why multi-layer transformers still overheat: magnetomotive force (MMF) builds linearly across m stacked non-interleaved layers, inducing eddy-current PROXIMITY losses proportional to m²! For sub-skin / moderate normalized thickness (Δ <= 1.25), Dowell’s expansion gives AC-to-DC Resistance Ratio Fr = R_ac / R_dc = 1 + ((5 * m^2 - 1) / 45) * (Δ^4)! Interleaving primary and secondary windings (P-S-P-S) resets MMF between sections, slashing effective m (e.g., from m = 8 down to m = 2) and eliminating ~94% of proximity loss!
Verify on a 70 kHz Non-Interleaved Foil Transformer—Foil_Thickness_h = 0.15 mm (B2), Copper_Skin_Depth_δ = 0.25 mm (B3 -> Normalized Thickness Δ = 0.15 / 0.25 = 0.6000 in E2, well below 1 skin depth!), NonInterleaved_Layers_m = 8 (B4 -> Dowell AC Factor Fr = 1 + ((5*64 - 1)/45)*(0.6^4) = 1.9187 in E3—nearly DOUBLE DC resistance!), Winding_DC_Resistance = 0.02 Ω (B5), and RMS_Current = 20.0 A (B6 -> DC Ohmic Loss = 400 * 0.02 = 8.00 W in E4): Actual AC Winding Copper Loss jumps to 8.00 * 1.91872 = 15.35 W (E5)!
| Dowell Proximity Metric (h=0.15 mm, δ=0.25 mm, m=8 Layers, R_dc=0.02 Ω, I=20 A) | Formula | Verified Output |
|---|---|---|
| Normalized Foil Thickness Δ | Foil_Thickness_h / Skin_Depth_δ | 0.6000 (< 1 Skin Depth!) |
| Dowell AC/DC Ratio Fr | 1 + ((5 * m^2 - 1) / 45) * (Δ^4) | 1.9187 (1.0547 Interleaved m=2) |
| DC Copper Loss P_dc (W) | (I_rms^2) * R_dc | 8.00 W |
| Total AC Winding Loss P_ac (W) | P_dc * Dowell_Fr | 15.35 W (8.44 W Interleaved m=2) |
Dowell Sub-Skin Proximity AC Winding Loss Formula
Validate Foil_Thickness > 0, Skin_Depth > 0, sub-skin asymptotic bound Δ = h / δ <= 1.25, Layer_Count m >= 1, R_dc > 0, and I_rms > 0!
=LET(th_h, B2, sk_d, B3, ly_m, B4, rd_c, B5, ir_a, B6, dl_r, th_h / sk_d, IF(OR(th_h <= 0, sk_d <= 0, dl_r > 1.25, ly_m < 1, rd_c <= 0, ir_a <= 0), "Invalid Sub-Skin Dowell Input (Delta > 1.25)", LET(fr_d, 1 + ((5 * (ly_m ^ 2) - 1) / 45) * (dl_r ^ 4), ROUND((ir_a ^ 2) * rd_c * fr_d, 2))))P-S-P-S Winding Interleaving Breakthrough Check (Set Effective Section Layers m = 2)
Interleave primary and secondary layers so effective MMF buildup drops from m = 8 to B4 = 2: Dowell Fr E3 plunges from 1.9187 to 1.0547, cutting Total AC Winding Loss E5 from 15.35 W down to 8.44 W using the exact same copper cross-section!
=IF(ISTEXT(E5), E5, ROUND(1 + ((5 * (B4 ^ 2) - 1) / 45) * ((B2 / B3) ^ 4), 4))Try the example
Use the original values or change them to test your understanding.