Planar Transformer: Dowell AC Resistance and iGSE Core Loss

For each harmonic, Ampère’s law gives the boundary MMF per ampere of primary current. In a primary layer the jump is the turns per layer; in a secondary layer it is the opposing secondary turns multiplied by the turns ratio. All layer terminals are series connected.

The Dowell slab solution is applied to each layer. With $z=(1+j)t\sqrt{\eta}/\delta$, the complex factors are $M=z\coth z$ and $D=z\tanh(z/2)$. A layer bounded by MMFs $F_L,F_R$ contributes $R_0[(F_R-F_L)^2M+2F_LF_RD]$, where $R_0=\rho\ell/(\eta wt)$ and $w$ is winding breadth. Real parts give resistance; imaginary parts divided by angular frequency give internal leakage inductance. Dielectric leakage is $\mu_0\ell/w\sum gF^2$. This is Dowell’s 1966 diffusion model; Chen et al., Appendix II, independently derive the same factors for planar layers.

Copper loss sums $I_{h,\mathrm{rms}}^2R_h$. The bipolar trapezoid is a square wave averaged over one transition interval, giving odd peak harmonics proportional to $\mathrm{sinc}(h\pi r)/h$; the RMS normalization is $1-4r/3$. No renormalization hides omitted harmonics.

The iGSE integrates $|dB/dt|^\alpha(\Delta B_{loop})^{\beta-\alpha}$ over each closed loop and divides the sum of loop energies by the full period. Each loop is partitioned from its global minimum to maximum and back. A work queue removes excursions from the rising and falling paths, then processes their nested loops. Crossing segments are split at their closure level, preserving their slope integrals. Extrema within 10⁻¹⁰ of the normalized amplitude use the largest following monotonic excursion as a deterministic, phase-independent tie-break. Analytical turning points are included for the third-harmonic family. The sinusoidal calibration uses the exact gamma-function cosine integral. PWM segments are integrated exactly; smooth flux is sampled piecewise linearly. See Venkatachalam et al. (2002), equations 9–13.

The N49 preset follows TDK’s June 2025 data at 100 °C over 500–1000 kHz and 25–50 mT peak. At 50 mT, a quadratic in log frequency passes through 80, 160 and 475 kW/m³ at 500, 700 and 1000 kHz; the endpoints are tabulated on page 2 and the middle point is read from the dashed 100 °C curve FAL0595-6 on page 5. The local frequency exponent is the slope of this curve. The flux exponent is ln(80/13.4)/ln(2), using the 25 mT/500 kHz point read from FAL0594-X. Reference loss and frequency move with the operating point; the iGSE integral uses those local exponents. These are fits made here, not manufacturer-supplied Steinmetz coefficients.

Validation excludes the 750 kHz/50 mT, 750 kHz/25 mT and 1 MHz/25 mT points from calibration. Their graph readings are approximately 193, 34.3 and 84.5 kW/m³. The separate E8.8 maximum-loss specification is an independent upper-bound check and is not a fit anchor.

Uniform series current within each winding; identical turn length and breadth for all layers; full ampere-turn cancellation at every harmonic. Copper has resistivity 1.724×10⁻⁸ Ω·m at 20 °C and resistance temperature coefficient 0.00393 K⁻¹ (rounded IACS values, NBS Handbook 100, 1966, sections 1.1a–b); thermal expansion is neglected. The copper-fill homogenization preserves DC conductance and uses effective conductivity ησ in the diffusion solution. Each layer has the entered copper thickness; gaps depend on whether adjacent layers belong to the same winding. No exterior-window field remains after ampere-turn cancellation.

Flux has zero DC offset and uniform effective core cross-section. The two chosen current and flux waveforms describe independent operating quantities; no converter circuit is solved. Material data must match core temperature. Layer numbers run from bottom to top; P means primary and S means secondary. The stack drawing shows layer order with exaggerated copper and dielectric spacing.

Dowell is a magnetoquasistatic, one-dimensional foil model. Narrow traces, radial current crowding, vias, unequal mean turn lengths, core-gap fringing, external fields, winding capacitance, and proximity to self-resonance require a richer model. Parallel layers and their current redistribution are outside this calculator. Leakage is the loaded, primary-referred differential value, including the AC internal contribution; it excludes magnetizing inductance.

The N49 preset is restricted to 500–1000 kHz and 25–50 mT peak at 100 °C; results are withheld outside this range until a custom fit is selected. Its held-out graphical checks agree within 10%, which is a screening tolerance rather than an accuracy guarantee for a particular core shape. The preset does not acquire a temperature correction when copper temperature changes. The iGSE does not capture DC-bias effects or loss relaxation during zero-voltage dwell; harmonic content outside the fit’s frequency band adds material uncertainty even at a valid fundamental.

Smooth flux uses 2048 uniform intervals by default, up to 4096, with analytical third-harmonic turning points added. The maximum keeps the full loss calculation and its phase/continuity diagnostics near the browser time budget; finer interval detail below this resolution is not resolved. PWM uses exact ramp and dwell knots. Current loss uses the entered highest odd harmonic; the next equally wide band is evaluated separately for both layouts, and the larger relative addition is reported as the convergence indicator. It is not a bound on the entire remaining tail. The harmonic bar chart shows the first 16 retained odd orders. Zero-range loops and excursions below 10⁻¹⁰ of the normalized waveform are omitted before evaluating a possibly negative β−α power. Positive α and β do not require β to exceed α. No thermal feedback or saturation dynamics are solved. Total loss means modeled copper plus core loss; it excludes dielectric, termination, shield, and converter losses.