Multilayer Rectangular Planar Windings

• MLRPWs are rectangular, multilayer PCB windings whose inductance depends solely on geometric parameters such as outer and inner dimensions, number of layers, and trace properties.
• The monomial-like inductance model is derived via multiple linear regression on a large FEM dataset, achieving high accuracy with mean error near 0% and MAE around 1.24%.
• This design approach enables rapid, geometry-only inductance estimation for high-frequency, high-power applications, supporting efficient design-space exploration and optimization.

Searching arXiv for the specified papers and closely related work on rectangular planar windings. Multilayer Rectangle-shaped Planar Windings (MLRPWs) are rectangle-shaped planar PCB windings that are folded or stacked into multiple layers and connected in series so that the magnetic flux generated by each layer has the same polarity. In the 2025 inductance-estimation literature, the term refers specifically to a geometry class targeted to high-frequency, high-power PCB windings, for which a compact empirical-inductive model was proposed in the form of a monomial-like closed-form inductance equation fitted by Multiple Linear Regression (MLR) to a large FEM-generated dataset of about 5,850 windings (Papadopoulos et al., 16 Jul 2025). Within that usage, MLRPWs are not merely rectangular variants of single-layer planar spirals: they are multilayer structures whose inductance depends on outer and inner rectangular dimensions, turns per layer, number of layers, trace width, turn spacing, and interlayer spacing.

1. Geometry class and nomenclature

An MLRPW is treated as a planar conductor structure whose inductance equation depends only on geometry. The winding is rectangle-shaped, possibly folded or stacked into multiple layers, and connected in series so that the magnetic flux generated by each layer has the same polarity (Papadopoulos et al., 16 Jul 2025). The estimated quantity is the winding’s overall inductance as obtained from electromagnetic simulation or impedance extraction, effectively the self-inductance of the complete series-connected winding structure. It is not presented as a leakage-inductance-only model.

The geometric variables are:

• $D_1$: outer dimension of the winding along one rectangle side
• $D_2$: outer dimension along the orthogonal rectangle side
• $d_1$: inner opening dimension corresponding to $D_1$
• $d_2$: inner opening dimension corresponding to $D_2$
• $\overline{D}_1 = (D_1+d_1)/2$: mean dimension in the $D_1$ direction
• $\overline{D}_2 = (D_2+d_2)/2$: mean dimension in the $D_2$ direction
• $D_2$0: copper trace width
• $D_2$1: spacing between adjacent turns on the same layer
• $D_2$2: number of turns per layer
• $D_2$3: number of layers
• $D_2$4: vertical spacing between two consecutive layers

For a given layer, the inner dimensions are not independent. The adopted relations are

$D_2$5

$D_2$6

Accordingly, once $D_2$7 are set, the inner window is fixed. The training set imposed the ordering restriction

$D_2$8

and the fitted exponents are asymmetric. The formula therefore must be used with the same convention.

The constant $D_2$9 shows that the fitted model scales with permeability of free space rather than with a specific magnetic-core reluctance model. The paper discusses keeping the aperture large enough to accommodate a central core leg such as EE or EI, but the equation itself is a geometric winding inductance estimator rather than a magnetic-core inductance formula. This suggests that the model is intended for rapid winding-level inductance estimation before any detailed core-coupled refinement.

2. Monomial-like inductance equations

The principal analytical result for MLRPWs is a monomial-like fitted form,

$d_1$0

with

$d_1$1

Using the reported regression coefficients, the fitted equation becomes

$d_1$2

where all geometric quantities are in SI units and $d_1$3 is in henries (Papadopoulos et al., 16 Jul 2025).

A simplified version was also reported:

$d_1$4

The simplification is motivated by the near-equality of the fitted exponents for $d_1$5 and $d_1$6, and by the negligible effect of $d_1$7. In both forms, the equation is geometry-only and closed-form. That combination is central to its intended role in design-space exploration and optimization.

The equation is explicitly empirical rather than a first-principles exact solution. Its significance lies in preserving monomial simplicity while extending inductance estimation to rectangular, multilayer planar windings, a geometry class for which direct reuse of classical single-layer square-coil formulas is less suitable.

3. Regression construction and simulation corpus

The fitted model was converted into a linear regression problem by taking base-10 logarithms. Starting from the monomial form, the logarithmic transform yields

$d_1$8

with

$d_1$9

This has the standard linear form $D_1$0. The coefficients were obtained by MLR from a full simulated dataset of 5,850 FEM samples, split randomly into 80% training and 20% evaluation subsets: 4,680 samples for training and 1,170 for evaluation. The splitting and fitting process was repeated several times to verify convergence and stability, and the fitted coefficients showed only small deviations around the reported nominal values (Papadopoulos et al., 16 Jul 2025).

The simulation corpus was generated from two simulated subsets:

• Subset A: $D_1$1, with $D_1$2
• Subset B: $D_1$3, with $D_1$4

A mixed-size subset $D_1$5, designated subset C, was not simulated to save computation time.

The remaining variables were sampled as follows:

• $D_1$6
• $D_1$7
• $D_1$8
• $D_1$9
• $d_2$0 for $d_2$1
• $d_2$2

Dataset A contained 1,800 samples, dataset B contained 4,050, and the total was 5,850 simulated windings. The reference inductance extracted from FEM is denoted $d_2$3, and the comparison metric is

$d_2$4

The paper states that the training data were generated by FEM simulations and that the simulation campaign took several months, but it does not explicitly name the software package or fully disclose mesh strategy, excitation syntax, or boundary-condition details in the provided text. The frequency interpretation is therefore quasi-static: the model is a geometry-based inductance estimator rather than a distributed parasitic model.

4. Quantitative accuracy and experimental validation

On the simulated evaluation data, the full equation achieved:

• mean error: $d_2$5
• standard deviation: $d_2$6
• mean absolute error: $d_2$7

The error histogram is described as approximately normal, and the equation was found not to be biased toward any $d_2$8 subgroup when samples with error $d_2$9 were grouped by number of layers (Papadopoulos et al., 16 Jul 2025).

For the simplified equation, the reported results are:

• mean error: $D_2$0
• standard deviation: $D_2$1
• mean absolute error: $D_2$2

Thus the simplified form causes only a small loss in accuracy.

Experimental validation was performed on ten windings. Nine samples were printed and folded in the laboratory, and one sample was manufactured by a commercial PCB vendor as a true multilayer PCB. Folding and custom fabrication were used to ensure equal interlayer spacing $D_2$3 in multilayer windings. For the commercial four-layer sample, where layer spacing was not perfectly uniform, $D_2$4 was represented by the mean of prepreg and core distances:

$D_2$5

The measurement method used a modified power amplifier as excitation source, with capability up to 30 V peak, 1.5 A peak, and 50 kHz. Winding impedance was determined from the excitation voltage and the amplitude and phase of the current response. Measurements were cross-checked with an HP-4284A high-precision LCR meter. The setup had sufficiently low parasitic capacitance to avoid measurable influence up to 200 kHz.

Among the ten measured practical windings, only one sample had about 6% error; all others were below that, many around 2–4% or better. The representative measured comparisons were:

Sample	Predicted vs measured	Error
#1	9.69 vs 9.38 µH	$D_2$6
#2	34.09 vs 33.51 µH	$D_2$7
#3	9.09 vs 8.91 µH	$D_2$8
#4	9.05 vs 8.53 µH	$D_2$9
#5	13.74 vs 13.48 µH	$\overline{D}_1 = (D_1+d_1)/2$0
#6	125.70 vs 120.80 µH	$\overline{D}_1 = (D_1+d_1)/2$1
#7	8.19 vs 8.26 µH	$\overline{D}_1 = (D_1+d_1)/2$2
#8	29.65 vs 30.87 µH	$\overline{D}_1 = (D_1+d_1)/2$3
#9	63.87 vs 66.40 µH	$\overline{D}_1 = (D_1+d_1)/2$4
#10	215.55 vs 222.57 µH	$\overline{D}_1 = (D_1+d_1)/2$5

Samples #7–#9 were inside the original dataset; the others had at least one parameter outside it. This suggests that interpolation within the covered ranges is strongly supported, whereas moderate extrapolation is empirically promising but not guaranteed.

5. Scaling behavior and design interpretation

The fitted dependence

$\overline{D}_1 = (D_1+d_1)/2$6

encodes several design trends (Papadopoulos et al., 16 Jul 2025).

First, the strongest positive geometric dependence is on the mean dimensions, $\overline{D}_1 = (D_1+d_1)/2$7. The paper relates this to current-sheet approximation reasoning: increasing mean perimeter increases positive mutual coupling among same-side traces and reduces the relative impact of opposing-side negative mutual inductance.

Second, both the number of turns per layer and the number of layers have strong exponents, $\overline{D}_1 = (D_1+d_1)/2$8 and $\overline{D}_1 = (D_1+d_1)/2$9. The practical consequence is that both adding turns and adding layers increase inductance roughly with a power close to 1.8. The similarity of these exponents also motivates the simplified composite variable $D_1$0.

Third, trace width has a mild negative exponent, $D_1$1. For fixed outer dimensions and turn count, wider traces shrink the inner window $D_1$2, bringing opposite-current segments closer. The stated design tradeoff is that larger $D_1$3 helps current and thermal performance, while smaller $D_1$4 helps inductance.

Fourth, the spacing exponent $D_1$5 is nearly zero. Within the studied range, spacing influences inductance only weakly, which is useful because spacing is often selected from insulation or voltage rules rather than from magnetic optimization.

Fifth, interlayer spacing contributes a weak negative factor, $D_1$6. The qualitative explanation given is that larger layer separation reduces mutual inductance and increases leakage flux between layers.

Although $D_1$7 and $D_1$8 themselves carry negative exponents, they also increase $D_1$9 and $\overline{D}_2 = (D_2+d_2)/2$0. The practical net effect is reported as larger outer size generally increasing inductance, especially when it enlarges mean turn dimensions.

The paper places these trends in a high-power framing. It selects $\overline{D}_2 = (D_2+d_2)/2$1 to $\overline{D}_2 = (D_2+d_2)/2$2 mm to correspond roughly to 5–10 A current levels with about $\overline{D}_2 = (D_2+d_2)/2$3 temperature rise, and $\overline{D}_2 = (D_2+d_2)/2$4 to $\overline{D}_2 = (D_2+d_2)/2$5 mm for up to 300 V turn-to-turn withstand with coating. That framing indicates that the model was built for practical PCB magnetics rather than for narrow RF-only layouts.

6. Relation to classical formulas and to single-layer rectangular generalization

The MLRPW model is positioned against older analytical formulas such as Wheeler-, Rosa-, and CSA-inspired forms, which are often developed for single-layer coils, are most natural for square or regular polygon geometries, are less adaptable to rectangular asymmetry, and are not straightforward to extend to multilayer structures with finite interlayer spacing (Papadopoulos et al., 16 Jul 2025).

A prior monomial formula for single-layer square-shaped windings was

$\overline{D}_2 = (D_2+d_2)/2$6

with

$\overline{D}_2 = (D_2+d_2)/2$7

The 2025 MLRPW equation generalizes the monomial approach by handling rectangular geometry explicitly through $\overline{D}_2 = (D_2+d_2)/2$8 or $\overline{D}_2 = (D_2+d_2)/2$9, and by handling multilayer windings explicitly through $D_2$0 and $D_2$1. In the square case, the new fit reduces to exponents

$D_2$2

which are reasonably close to the original single-layer-square monomial exponents. This was presented as evidence that the new formula is a natural extension rather than a disconnected empirical fit.

A distinct 2025 line of work generalized square-planar single-layer inductance formulas to rectangular single-layer windings by replacing the square outer-side parameter with a Power Mean of the two outer dimensions (Papadopoulos et al., 22 Jul 2025). That work examined Wheeler, Rosa, and Monomial formulas on 2,633 rectangle-shaped single-layer windings and recommended $D_2$3 for Wheeler and Rosa and $D_2$4 for Monomial. However, it explicitly did not develop a direct inductance formula for MLRPWs; it treated only coreless, single-layer, printed rectangular planar windings. For MLRPWs, it is therefore not directly applicable as published. The contrast is methodologically important: Power Mean substitution addresses shape equivalence in one planar layer, whereas the MLRPW model explicitly encodes multilayer count and interlayer spacing.

A common misconception is that multilayer planar inductance can be estimated adequately by replacing the turn count $D_2$5 in a single-layer equation with total turns $D_2$6. The MLRPW paper rejects that as a general rule for higher $D_2$7, noting that leakage flux increases, interlayer mutual coupling decreases as layer distance increases, and interlayer spacing must explicitly enter the model.

7. Applicability limits and unresolved issues

The most reliable operating region is the trained geometry range:

• $D_2$8 from 70 to 160 mm in the subset pattern used
• $D_2$9–5 mm
• $D_2$00–0.5 mm
• $D_2$01
• $D_2$02–4
• $D_2$03–1.5 mm
• $D_2$04 mm

The paper does not claim formal extrapolation validity outside these ranges, even though several laboratory samples with one or more parameters outside the training set still showed good agreement (Papadopoulos et al., 16 Jul 2025). The caveat matters especially for $D_2$05 or $D_2$06, $D_2$07, very different $D_2$08, mixed $D_2$09 combinations omitted from simulation, and much smaller or larger traces or windows than those trained.

The model is empirical and quasi-static. It is not intended to replace full electromagnetic simulation when geometry is highly irregular, dielectric or copper stack-up is nonuniform in a way not captured by an effective $D_2$10, frequency-dependent parasitics become important, or magnetic cores or nearby conductors substantially modify the field. It does not model AC resistance, proximity or skin-effect losses, self-capacitance, self-resonant frequency, or frequency-dependent inductance roll-off.

A further practical limitation is that real multilayer PCBs may not have uniform interlayer spacing. The reported handling of the commercial four-layer sample by using a mean $D_2$11 shows one practical workaround, but it also indicates that a single spacing parameter is an approximation when the stack-up is nonuniform.

The omission of mixed-size subset C from the training data is also a methodological caveat. The formula worked well there in practice, but that region was not directly included in fitting. A plausible implication is that the fitted law captures broad geometric regularities rather than only memorizing the simulated grid, but the paper does not convert that observation into a formal extrapolation guarantee.

In summary, MLRPWs in the 2025 arXiv literature denote a specific multilayer rectangular PCB winding class for which a closed-form, FEM-calibrated, geometry-only inductance law was established. Its main technical importance lies in combining rectangular asymmetry, multilayer count, and interlayer spacing within a compact monomial structure, while maintaining evaluation accuracy of $D_2$12, $D_2$13, and $D_2$14 on the simulated evaluation set and good agreement with measured hardware, including several out-of-range cases (Papadopoulos et al., 16 Jul 2025).