What five Ansys Maxwell 2D FEA simulations across 2-layer to 6-layer PCB winding geometries, spanning EQ30-class to ER40-class cores, reveal about the limits and reliability of the analytical approach — and a free calculator to use it in your own designs.
Planar transformers built on PCB substrates have become the preferred magnetic component in high-density power converters — LLC resonant converters, dual active bridge (DAB) topologies, and isolated DC-DC stages in EV onboard chargers. The low profile, excellent thermal behaviour, and manufacturing repeatability make them compelling. But one parameter that defines much of their electrical behaviour in circuit is often estimated loosely or pulled from FEA late in the design cycle: leakage inductance.
In an LLC resonant converter, the leakage inductance of the transformer is the resonant inductor. The converter’s gain curve, ZVS range, and operating frequency all depend on it directly. In a DAB converter, leakage inductance sets the power transfer characteristic. In any isolated topology operating at high switching frequencies, leakage inductance determines the magnitude of voltage spikes at commutation and the energy available to drive resonant transitions.
Getting the number right at the design stage — before the PCB is fabricated — is therefore worth the effort. This post documents what I found when I compared a standard analytical formula against Ansys Maxwell 2D FEA simulations across five PCB winding geometries, ranging from a simple 2-layer non-interleaved structure on an EQ30-class core to a 6-layer triple-interleaved stack on a larger ER40-class core.
The 1D Analytical Formula
The standard analytical expression for leakage inductance in a multilayer planar transformer winding is:
L_leak_1D = m × (μ₀ × lᵤ / w_eff) × [2h_cu/3 + h_ins] × K_R where: m = number of primary-secondary interleaving boundaries μ₀ = 4π × 10⁻⁷ H/m (permeability of free space) lᵤ = mean turn length (metres) w_eff = effective winding breadth (metres) — see below h_cu = copper layer thickness (μm, converted to metres) h_ins = insulation thickness between layers (μm, converted to metres) K_R = Rogowski factor — accounts for vertical flux compression |
The formula rests on two physical assumptions. First, that the magnetic field in the transformer window is uniform across the winding breadth — a 1D approximation that ignores fringing at the trace edges. Second, that the winding stack fills the window height uniformly — which breaks down for sparse 2-layer designs in tall cores.
Two correction terms extend the basic formula to handle real geometries. The effective winding breadth w_eff captures horizontal fringing at trace edges using a logarithmic correction derived from the window geometry and insulation thickness:
w_eff = w_trace + (h_ins ÷ π) × ln[ 1 + π × (w_window − w_trace) ÷ h_ins ] w_eff is capped at w_window — it cannot exceed the physical window width. |
The Rogowski factor K_R captures the fact that flux lines at the top and bottom of the winding stack bow outward rather than remaining horizontal, reducing the effective stored energy. Cross-checking against the FEA data across five geometries showed that expressing K_R as a function of the trace width to stack height ratio — rather than the insulation thickness to stack height ratio used in the initial derivation — produced a tighter and more consistent fit:
ξ = π × w_trace ÷ H_stack K_R = 1 − (1 − e⁻ᵖ) ÷ ξ K_R approaches 1.0 as the trace width shrinks relative to the stack height. |
The full leakage inductance expression also includes the cumulative turns count. Leakage inductance scales with the square of the total turns per winding (N²), divided by the interleaving boundary count m:
L_leak_1D = (Np² ÷ m) × (μ₀ × lᵤ / w_eff) × [2h_cu/3 + h_ins] × K_R Validated to date for designs where N_p = m (turns-per-winding equal to interleaving boundary count). Configurations where N_p ≠ m should be treated as analytical estimates pending further validation. |
Figure-1: MMF profile across the window height for non-interleaved (m=1) vs interleaved (m=2, m=3) winding stacks
The Role of Winding Interleaving
The parameter m — the number of primary-secondary interfaces in the winding stack — has a direct and powerful effect on leakage inductance. The mechanism is the MMF profile across the window height.
In a non-interleaved arrangement (all primary turns together, all secondary turns together), the MMF builds from zero at the bottom, rises to a peak at the primary-secondary boundary, then falls back to zero at the top. The area under the MMF² profile is large, which means large stored magnetic energy in the window — and therefore large leakage inductance.
Each additional interleaving boundary splits this profile into smaller humps. For m interleaving boundaries, the peak MMF is suppressed and the total stored energy — and therefore the leakage inductance — scales approximately as 1/m². This is visible directly in the formula: L_leak_1D is proportional to m, not m², because the turns count in the denominator also changes with the winding arrangement. The practical result is that moving from m=1 to m=3 reduces leakage inductance by roughly a factor of 3 to 4 depending on geometry.
Interleaving boundary count ‘m’ is the most powerful single design variable for controlling leakage inductance in a planar transformer. Doubling m approximately halves the leakage inductance — without changing the core, the copper weight, or the turns count.
Extracting Leakage Inductance from FEA — The L-Matrix Method
Leakage inductance cannot be read directly from an Ansys Maxwell 2D simulation as a single output. The correct extraction method uses the inductance matrix from a magnetostatic or eddy current solution with opposing winding excitation.
The simulation applies +1A to the primary winding and −1A to the secondary (scaled for turns ratio), then extracts the full L-matrix: L11 (primary self-inductance), L22 (secondary self-inductance referred to primary), and L12 (mutual inductance). The total leakage inductance referred to the primary is then:
L_leak = L11 + L22 − (2 × L12) This extracts the non-coupled flux energy from both windings. The result is independent of magnetising inductance and therefore independent of core air gap — a property the analytical formula shares. |
Figure-2: Ansys Maxwell 2D simulation setup showing winding excitation and field solution for one of the five validation cases
Five Validation Cases — Geometry and Results
The five cases span a range of winding geometries — from a minimal 2-layer structure to a 6-layer interleaved stack, from a narrow EQ30-class window to a larger ER40-class window roughly double the area. All cases use copper traces on FR4 PCB substrate with a standard EQ-class, ELP-class, or ER-class planar core.
| Case | Layers | m | w_window | H_window | MTL | h_cu | h_ins | Trace width |
| 1 | 2 (P-S) | 1 | 10 mm | 14 mm | 52 mm | 140 μm | 200 μm | 8.5 mm |
| 2 | 4 (P-S-P-S) | 2 | 10 mm | 14 mm | 52 mm | 140 μm | 200 μm | 8.5 mm |
| 3 | 2 (P-S) | 1 | 12 mm | 16 mm | 60 mm | 105 μm | 300 μm | 6.0 mm |
| 4 | 6 (P-S-P-S-P-S) | 3 | 15 mm | 20 mm | 70 mm | 105 μm | 200 μm | 13.0 mm |
| 5 | 6 (P-S-P-S-P-S) | 3 | 18 mm | 25 mm | 95 mm | 105 μm | 250 μm | 16.0 mm |
For each case I ran the Ansys Maxwell 2D L-matrix extraction and recorded L11, L22, and L12. The leakage inductance was computed using the formula above. I then ran the same geometry through the analytical formula and compared results.
| Case | L11 (μH) | L22 (μH) | L12 (μH) | FEA L_leak | Tool L_leak_1D | Error |
| 1 | 4.1266 | 4.1259 | 4.1252 | 2.1 nH | 2.163 nH | +3.0% |
| 2 | 16.379 | 16.381 | 16.378 | 4.0 nH | 4.213 nH | +5.3% |
| 3 | 4.1593 | 4.1581 | 4.1566 | 4.2 nH | 4.243 nH | +1.0% |
| 4 | 35.873 | 35.868 | 35.868 | 5.0 nH | 5.174 nH | +3.5% |
| 5 | 55.922 | 55.913 | 55.914 | 7.0 nH | 6.785 nH | −3.1% |
Figure-3: Bar chart comparing analytical L_leak_1D vs Ansys FEA result across all five cases
Observations from the Results
Three patterns emerge from the five cases that I think are worth noting.
Observation 1 — The corrected formula holds within a tight band across very different geometries
The five cases span window areas from 140 mm² (Case 1) to 450 mm² (Case 5) — roughly a 3× range — and layer counts from 2 to 6. Across all of that, the analytical result stayed within +1.0% to +5.3% of the Ansys value for four of the five cases, with Case 5 landing at −3.1%. That is a meaningfully tighter spread than I expected going in, and it suggests the corrected w_eff, K_R, and turns-scaling terms are capturing the dominant physics rather than fitting to a narrow geometry range.
Observation 2 — The error sign is no longer one-directional
Cases 1 through 4 all came in slightly above the Ansys value, while Case 5 — the largest core in the set — came in slightly below. A formula with a consistent one-directional bias usually signals a missing physical term; a formula whose error crosses zero across a reasonably wide geometry sweep is a better sign that the residual error is closer to genuine model uncertainty than to a systematic gap. I would not over-read five data points, but this is encouraging.
Observation 3 — The turns-squared scaling needs its own validation envelope
Leakage inductance scales with the square of the turns count per winding, divided by the interleaving boundary count (Np² / m). In every case validated so far, the turns count per winding happened to equal the number of interleaving boundaries (Np = m). That is a coincidence of how the test matrix was built, not a general property — a non-interleaved design with five primary turns, for example, has not yet been checked against FEA. I am treating the Np² scaling as theoretically sound but only empirically confirmed for Np = m configurations, and I would not extrapolate confidently beyond that until further cases are run.
Figure-4: Scatter plot of analytical error vs window area, with data points labelled by case number
Across five validated geometries spanning roughly EQ30-class to ER40-class cores, the corrected analytical formula holds within ±10% of Ansys Maxwell 2D FEA in every case — with observed error ranging from −3.1% to +5.3%.
This is the working tolerance I am comfortable publishing for the calculator. It is not a guarantee for every possible geometry, and the Np²/m turns scaling specifically has only been confirmed where turns count equals the interleaving boundary count.
What Is Still Open
Five validated cases is a reasonable starting dataset, but it is not exhaustive, and I want to be specific about what has and has not been checked.
Every case in this set used Np = m — the turns count per winding happened to equal the interleaving boundary count in all five geometries. That was a natural consequence of how the test matrix was built (more interleaving boundaries came with more layers, and more layers came with more turns), but it means the Np² scaling in the formula has not been independently isolated. A non-interleaved 2-layer design with five turns per winding, for instance, predicts a leakage inductance 25× higher than the same geometry with one turn — that specific scaling has not yet been checked against FEA.
All five cases also used a single insulation thickness value uniformly through the stack, and all used rectangular FR4-substrate copper traces of broadly similar aspect ratio. Designs with mixed insulation thicknesses between different layer pairs, non-rectangular trace profiles, or asymmetric primary/secondary trace widths fall outside what has been tested.
Working accuracy guide for the corrected 1D analytical formula, based on five validated cases:
|
Free Calculator — Planar Transformer Leakage Inductance
I built an Excel-based calculator that implements the formula above with the Rogowski and fringing corrections, the fill factor checks, and an empirical correction term fitted to the FEA data. It is free to download from circuitbrilliance.com.
The calculator has four sheets: an Index sheet with scope and disclaimer, an Inputs sheet where you enter core geometry and winding stack layer by layer, a Process sheet showing every calculation step transparently, and a Report sheet with the final result and accuracy assessment.
The tool reports the base analytical result, L_leak_1D, along with the intermediate fringing, Rogowski, and turns-scaling terms shown transparently on a dedicated Process sheet. Based on five validated geometries spanning roughly EQ30-class to ER40-class cores, the tool carries a working accuracy tolerance of ±10%, with observed error in the validated cases ranging from −3.1% to +5.3%.
Closing Note
The 1D analytical formula for planar transformer leakage inductance, with the fringing, Rogowski, and turns-scaling corrections described here, held within ±10% of Ansys Maxwell 2D FEA across five geometries spanning roughly a 3× range of core window area. That is a useful working tolerance for early-stage design — fast enough to iterate through several winding stack options before committing to FEA, and close enough to trust the relative comparison between options.
Getting to this point took more than one pass. An earlier version of the formula — using a different Rogowski factor and missing the turns-squared scaling entirely — produced errors as large as −25% on the same first four geometries before the correction. The five-case dataset here reflects the corrected formula, cross-checked independently by hand calculation against both the spreadsheet and a prototype web implementation before being trusted.
I will continue extending the validation set, particularly for configurations where the turns count per winding does not equal the interleaving boundary count, since that combination has not yet been tested against FEA. If you run the calculator on your own geometry and get results that diverge meaningfully from measurement or FEA, I would be interested to hear the geometry — particularly outside the Np = m configuration, where the formula is least proven so far.