Magnetics · Appendix A — Area product by magnetic kind (draft)

Draft appendix to the magnetics lesson, which the 2025 lectures V8 (inductor design) and V9 (transformer design) cover until the 2026 lesson is written. Numbers come from scripts/area_product_bias.py and are recorded in area-product-checks.json.

V8 sizes an inductor from the stored energy and the window. V9 sizes a transformer by iteration, from the balance of copper loss and core loss. Both methods give an area product $A_p = W_aA_e$. This appendix shows why one flux density cannot serve both.

A size proxy that uses one flux density for every magnetic gives a biased result. It sizes DC-dominated chokes well, but it makes AC-flux magnetics far too small. An audit of six built components showed this pattern. The DC chokes landed within −30 % to +42 % of the real area product. The AC-flux magnetics (an ICT, an ICMCI and a DAB transformer) came out 64 % to 74 % too small.

This appendix explains the cause and gives the fix. Saturation limits the flux in a choke. Core loss limits the flux in an AC-flux magnetic. The two kinds therefore need two sizing methods: the core-geometry constant $K_g$ for chokes, and $K_{gfe}$ for AC-flux parts.

A.1 What the area product says

A magnetic must satisfy two conditions at the same time. The core must carry the flux without exceeding a flux density limit $\hat B$. The window must hold the copper at a current density $J$.

The peak flux linkage per turn $\hat\lambda$ sets the core area. The sum of winding ampere-turns, divided by the turns of one reference winding, sets the copper area:

Here $A_e$ is the effective core area, $W_a$ the window area, $K_u$ the fraction of the window that holds copper, and $I_{\mathrm{tot}}=\sum_j (N_j/N)I_{j,\mathrm{rms}}$. Multiply the two conditions and the turns count $N$ cancels:

This one equation covers both kinds. For a choke, $\hat\lambda = L\hat I$ and $I_{\mathrm{tot}}=I_{\mathrm{rms}}$, so $A_p = L\hat I I_{\mathrm{rms}}/(K_uJ\hat B)$. For a transformer with a square-wave voltage $V$ at frequency $f$, $\hat\lambda = V/(4f)$. The result is the familiar $A_p = P_t/(4K_uJ\hat Bf)$, with the apparent power $P_t=\sum V_jI_j$.

The equation is exact for a full window. If a design fills its window, the equality holds with the real $K_u$, $J$ and $\hat B$ of that design. A proxy that inserts fixed values $K_{u0}$, $J_0$ and $B_0$ is therefore wrong by exactly

The bias has no other source. To explain it, find which of the three factors moves, and why.

A.2 Which limit sets the flux density

In a DC choke, the DC current sets most of the flux. The ripple adds only a small AC swing. In the buck choke of section A.5, the peak flux density is 300 mT but the AC amplitude is only 27 mT. Core loss is small, so saturation sets $\hat B$. The design value is about $0.8B_{\mathrm{sat}}$.

In an AC-flux magnetic, all of the flux alternates. The core loss per volume follows the Steinmetz equation:

The Ferroxcube 3C90 datasheet gives three loss points at 100 °C:

Two pairs of points give the exponents directly, without a curve fit:

At saturation flux, the core cannot get rid of its heat. At 100 kHz and $\hat B$ = 300 mT, 3C90 loses 1.24 MW/m³. An ETD54 core (35.5 cm³) would dissipate 44 W. The same core can lose only about 8 W at a 40 K temperature rise. Core loss, and not saturation, sets $\hat B$ in this part. The usable value is near 100 mT, a third of the saturation value.

This is the main term of the bias. A proxy that uses $B_0$ = 300 mT for an AC-flux part overstates $\hat B$ by about three times. It understates $A_p$ by the same factor.

The window fill adds a second term. A DC choke uses solid round wire, and $K_u$ = 0.4 is a normal value (McLyman, ch. 4). An AC-flux part at 100 kHz needs litz wire or foil to control skin and proximity loss. Litz strands, serving and air between bundles reduce $K_u$ to about 0.25 to 0.30. A transformer with reinforced isolation also needs tape, margins or triple-insulated wire, and $K_u$ falls to about 0.2.

A.3 DC-dominated chokes: Kg at saturation flux

The core-geometry method (Erickson and Maksimović, ch. 14) replaces the fixed current density with a copper-loss budget $P_{cu}$. The winding resistance is $R = \rho N^2\,\mathrm{MLT}/(K_uW_a)$, where MLT is the mean length of a turn. Combine this with $N = L\hat I/(\hat BA_e)$:

$K_g$ depends only on the core. The right side depends only on the specification. A core is large enough if its $K_g$ is above the requirement. For a choke, $\hat B$ is the saturation design limit.

Worked example: buck choke 48 V to 12 V, 30 A, 100 kHz. The specification is $L$ = 15 µH, $\Delta I$ = 6 A, $\hat I$ = 33 A and $I_{\mathrm{rms}}$ = 30.05 A. Use $\hat B$ = 0.30 T, $K_u$ = 0.40, and copper at 100 °C with $\rho$ = 2.26 × 10⁻⁸ Ω m. The loss budget is the loss that gives a 40 K rise on the core in question (section A.5).

ETD39 has $A_e$ = 125 mm². The turns count is $N = L\hat I/(\hat BA_e)$ = (15 µH × 33 A)/(0.30 T × 125 mm²) = 13.2, so wind 14 turns. The core loss is 0.04 W, which confirms that the choke is DC-dominated.

Mueller (2025) uses the same logic for powder-core DC filter inductors. He prefers $K_g$ to $A_p$, because $K_g$ sets a loss and not an assumed current density. A powder core has no gap to adjust, so a material "saturation constant" $\Gamma = B_{\max}^2$ replaces $\hat B^2$. The method ignores AC core loss by assumption. That assumption is the boundary of this appendix: $K_g$ at saturation is valid only while the AC flux is small.

A.4 AC-flux magnetics: Kgfe

For an AC-flux part, the flux density is a free design variable. More turns lower $\hat B$ and the core loss, but they raise the copper loss:

Here $K_{fe} = kf^{\alpha}$ is the Steinmetz coefficient at the operating frequency. The quantity $\lambda_1$ is the peak-to-peak flux linkage per turn, so $\hat B = \lambda_1/(2NA_e)$. Set $dP_{\mathrm{tot}}/d\hat B = 0$. The total loss is a minimum when

For 3C90, $\beta$ = 2.49, so core loss is 0.80 of copper loss at the optimum. Insert the optimum into $P_{\mathrm{tot}} \le P_{\max}$. This gives the condition of Erickson and Maksimović (ch. 15):

No flux density appears in this condition. The method picks $\hat B$ from the loss balance at the operating frequency. It does not assume $\hat B$. The script checks the closed form against a numerical loss minimum on all seven ETD cores, with agreement to 10⁻⁴.

Worked example: DAB transformer, 400 V, 1.5 kW, 100 kHz. The winding voltage is a ±400 V square wave, so $\lambda_1 = V/(2f)$ = 2.0 mV s. The turns ratio is 1:1 and each winding carries 4.5 A rms, so $I_{\mathrm{tot}}$ = 9.0 A. Use $K_u$ = 0.20 for litz wire with reinforced isolation. In SI units, $K_{fe}$ = 2.48 × 10⁷ W m⁻³ T⁻β.

$K_{gfe}$ has a unit that depends on $\beta$. Compare it only with a requirement in the same SI units. On ETD54, the loss-optimal design has 39 turns and $\hat B$ = 92 mT. The losses are $P_{fe}$ = 2.31 W and $P_{cu}$ = 2.88 W, which is 5.2 W against an 8.0 W limit.

Kazimierczuk and Sekiya (2009) give area-product equations for resonant inductors in terms of the loaded quality factor, the output power and the frequency. Their equations have the correct form, but $\hat B$ and $K_u$ are still inputs. Their examples compare a single-wire and a multiple-strand winding, with skin, proximity and fringing effects. $K_{gfe}$ removes the guess of $\hat B$, which is the larger of the two errors.

A.5 Teaching example: six magnetics on one core family

The example sizes six magnetics on Ferroxcube ETD cores in 3C90 at 100 kHz and 100 °C. All core and material data come from the Ferroxcube datasheets. Nothing is fitted to the result. Each part is sized three times:

1. Old proxy. $A_p = \hat\lambda I_{\mathrm{tot}}/(K_{u0}J_0B_0)$, with $K_{u0}$ = 0.40, $J_0$ = 4 A/mm² and $B_0$ = 0.30 T for every kind.
2. Method by kind. DC chokes use $K_g$ at $\hat B$ = 0.30 T. AC-flux parts use $K_{gfe}$. Each method selects the smallest ETD core that meets its condition.
3. Reference. For each core, a numerical search finds the turns count with the lowest total loss and $\hat B \le$ 0.30 T. The reference is the smallest core whose loss gives a temperature rise of 40 K or less.

The temperature limit uses McLyman's natural-convection relation $\Delta T = 450\,(P/A_t)^{0.826}$, with $P$ in W and $A_t$ in cm². $A_t$ is the surface of a box around the core and the winding. ETD29 can lose 2.5 W and ETD59 can lose 9.8 W.

The example shows the same pattern as the audit. The DC chokes land between +6 % and +45 %. The AC-flux parts land between −71 % and −85 %.

The figure shows where the error comes from. For the DC chokes, $\hat B$ and $K_u$ match the proxy. Only $J$ moves, because a small core can shed more heat per unit of copper. For the AC-flux parts, the flux factor is 0.31 to 0.38 and the fill factor is 0.50 to 0.75. The current density stays within 5 % of $J_0$. The audit hypothesis named thermally limited $B$ and $J$ plus litz and isolation $K_u$. This example confirms $B$ and $K_u$. It does not confirm $J$.

Bias between kinds. Take the geometric mean of $A_{p,\mathrm{proxy}}/A_{p,\mathrm{ref}}$ for each kind. It is 1.21 for the DC chokes and 0.22 for the AC-flux parts. The bias between kinds is the ratio of the two, 5.6×. The method by kind selects the reference core for all six parts, so its bias is 1.00×.

That 1.00× does not validate the method. The reference uses the same loss model as $K_{gfe}$, so agreement is guaranteed. The example shows the mechanism of the bias. Only measured parts can show that the method is correct.

A.6 For KPI owners: the acceptance rule

An area-product KPI that uses one flux density for all magnetics compares parts of different kinds on a biased scale. The fix has three rules:

1. Report the area product separately for each magnetic kind.
2. For DC-dominated chokes, keep $A_p$ or $K_g$ at the saturation flux density.
3. For AC-flux magnetics (ICT, coupled inductors, transformers), use $K_{gfe}$. It sets the flux swing from core loss at the operating frequency.

$K_g$ alone does not remove the bias, because it still sizes the flux at saturation.

Acceptance test. Replay the six components of the earlier proxy audit with the new method. Fit nothing to those six parts. Calculate the bias between kinds as in section A.5. If the bias is below 1.5×, the KPI is qualified. If it is 1.5× or more, report the KPI as unqualified.

The audit gave a bias of 3× to 4× with the old proxy. This appendix predicts that most of the bias comes from $\hat B$, with a smaller part from $K_u$. If the replay still fails, check $K_u$ first. Litz construction and isolation build-up differ more from part to part than the core loss does.

A.7 What if

• What if the core size steps? The reference rejects ETD49 for the DAB transformer by 0.3 % of loss (6.47 W against 6.45 W). With a small fan, or with 3 % less current, ETD49 is sufficient and the proxy error becomes −74 %. Catalog steps of 1.5× to 1.8× in $A_p$ limit the resolution of any single-part comparison.
• What if the voltage is a square wave? Steinmetz data come from sine-wave tests. Square-wave flux with short transitions has more loss. The improved generalized Steinmetz equation (iGSE) corrects this by using only the same three parameters (Venkatachalam et al., 2002).
• What if a choke has a large ripple? A boundary-mode choke has $\Delta I = 2I$, so its AC flux is as large as its DC flux. Its core loss is not small. Size it with both $K_g$ and $K_{gfe}$, and use the larger core.
• What if the DC flux is not zero? A DC bias raises ferrite core loss at the same AC amplitude. An ICT with unequal phase currents has a DC flux, so check its saturation margin as well.
• What if a powder core is used? Powder cores have a soft saturation and higher core loss than ferrite. Use Mueller's $\Gamma$ method for a DC-dominated powder choke. For AC flux, use $K_{gfe}$ with the Steinmetz data of the powder material.

Sources

• R. W. Erickson and D. Maksimović, Fundamentals of Power Electronics, 2nd ed., Kluwer, 2001: ch. 14 (inductor design, $K_g$), ch. 15 (transformer design, $K_{gfe}$).
• C. W. T. McLyman, Transformer and Inductor Design Handbook, 4th ed., CRC Press, 2011: ch. 4 (window utilization), ch. 6 (temperature rise).
• M. K. Kazimierczuk and H. Sekiya, "Design of AC resonant inductors using area product method," IEEE ECCE 2009, pp. 994–1001, doi:10.1109/ECCE.2009.5316501.
• L. Mueller, "Using the Area Product or Core Geometry Method in DC Filter Inductor design with powder cores," LinkedIn, 6 January 2025, link.
• K. Venkatachalam, C. R. Sullivan, T. Abdallah and H. Tacca, "Accurate prediction of ferrite core loss with nonsinusoidal waveforms using only Steinmetz parameters," IEEE COMPEL 2002.
• Ferroxcube, 3C90 material specification, 2008, and ETD29/16/10 … ETD59/31/22 product specifications, 2016.

Reproduce

Use the repository environment ntbees2:

python scripts/area_product_bias.py

The script writes area-product-checks.json and figures/area-product-bias.svg. It asserts three things:

1. The Steinmetz parameters return all three datasheet points.
2. The closed-form $K_{gfe}$ equals the numerical loss minimum.
3. The three proxy factors multiply to the proxy error.