What is a planar transformer?

A transformer whose windings are PCB traces or thin copper sheets rather than wire, sandwiched in a flat ferrite core. It's low-profile, highly repeatable because winding geometry is precise, and easy to cool. It's used in power supplies running at hundreds of kHz and above, including automotive.

Does thicker copper reduce resistance?

Not necessarily at high frequency. At 500 kHz current crowds into about 93 μm from the surface, and proximity effect from neighbouring layers' fields skews it further. With four primary layers stacked separately from four secondary layers, 70 μm copper has 1.55 times DC resistance but 140 μm has 8.4 times, so actual resistance rose about 2.7-fold even though DC resistance halved.

What does winding order change?

Interleaving primary and secondary layers reduces the field (MMF) between layers. For four layers each, peak MMF fell from 4 to 1, AC resistance at 70 μm from 1.55 to 1.03 times DC, and the field energy corresponding to leakage inductance to about one-eleventh. Inter-winding capacitance increases in return.

Planar Transformer

Category: Electromagnetic Analysis – Transformers | Revised 2026-10-01

Planar Transformers

Theory: skin and proximity effects

Overview

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Our planar transformer windings run hotter than expected. Will thicker copper fix it?

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At high frequency, current flows only near the conductor surface (skin effect). With many stacked layers, other layers' fields push current to one side of each conductor (proximity effect). If all primary layers are stacked before the secondaries, the field between layers grows with each added layer. Thick copper in that field worsens current crowding, and resistance can even rise when copper is thickened. So first reduce inter-layer field by winding order, then choose a thickness suited to the skin depth. Dowell's equation captures all this.

Dowell's equation

$$ F_R = \frac{R_{ac}}{R_{dc}} = \Delta\left[\frac{\sinh 2\Delta + \sin 2\Delta}{\cosh 2\Delta - \cos 2\Delta} + \frac{2(m^2 - 1)}{3}\,\frac{\sinh\Delta - \sin\Delta}{\cosh\Delta + \cos\Delta}\right],\qquad \Delta = \frac{h}{\delta} $$

h is copper thickness, δ skin depth and m the number of layers over which MMF rises from zero to its peak (m = 4 for four primary layers stacked before four secondaries, m = 1 when interleaved).

Skin depth and leakage-field energy

$$ \delta = \sqrt{\frac{\rho}{\pi f \mu_0}},\qquad E_{leak} \propto \sum_{gaps} \mathcal{F}^2 $$

ρ is copper resistivity (1.72×10⁻⁸ Ω·m). Leakage inductance is taken as proportional to the sum of squared inter-layer MMF F.

Coffee Break Trivia

Dowell's 1966 paper

In 1966 Dowell treated transformer winding layers as flat sheets and derived an AC-resistance formula combining skin and proximity effects. It's also applied to round-wire windings by substituting sheets of equal cross-section, and underpins high-frequency transformer and inductor design. Planar windings are already flat copper, so they fit Dowell's assumptions well; where layers don't fill the core window width, or at layer edges, 2D effects appear and finite elements are used for precision.

Worked examples

Example 1: AC-to-DC resistance ratio F_R (500 kHz, skin depth 93 μm)

Copper thickness	h/δ	Separated (m = 4)	Half interleaved (m = 2)	Interleaved (m = 1)
35 μm	0.37	1.03	1.01	1.00
70 μm	0.75	1.55	1.13	1.03
140 μm	1.50	8.38	2.78	1.38

At 1 MHz (skin depth 66 μm): 3.11 / 1.51 / 1.11 for 70 μm and 20.7 / 5.77 / 2.04 for 140 μm. At 100 kHz (209 μm), even 140 μm stays at 1.35 / 1.08 / 1.02.

Example 2: winding order and leakage field (4 primary, 4 secondary layers)

Order	Peak MMF	Leakage-field energy (relative)
PPPPSSSS	4	1.000
PPSSPPSS	2	0.273
PSPSPSPS	1	0.091

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At 140 μm DC resistance halves, yet in the separated arrangement actual resistance increases.

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Actual resistance is DC resistance times F_R, so both the thickness-inverse DC resistance and F_R matter. Relative to 70 μm, separated windings at 140 μm rise by 8.38/2 ÷ 1.55 ≈ 2.7 times. Interleaved, F_R stays at 1.38, so thicker copper cuts resistance to 1.38/2 ÷ 1.03 ≈ 0.67 times. Whether thick copper helps is decided by winding order. Interleaving also cuts leakage inductance to about one-eleventh, reducing switching voltage spikes, but bringing primary and secondary closer raises inter-winding capacitance and common-mode noise paths. Resonant converters sometimes use leakage inductance as a circuit element, so choose the order to suit the purpose.

Design and analysis workflow

Compute skin depth at operating frequency and shortlist copper thicknesses.
Sketch MMF distribution for each winding order to set m.
Compute AC resistance with Dowell's equation and sum losses over current harmonics.
Check losses near layer edges and core gaps with 2D/3D finite elements.
Measure leakage inductance and inter-winding capacitance against circuit requirements.

Coffee Break Trivia

“Thicker copper ran hotter”

To cool a planar transformer in a 500 kHz supply, PCB copper was increased from 70 to 140 μm, but winding temperature actually rose. The windings had four primary layers above four secondary layers. Dowell's equation showed AC resistance had grown about 2.7-fold. Switching to an interleaved arrangement brought 140 μm resistance to about two-thirds of the 70 μm value, and temperature fell below target.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Loss from DC resistance	Loss underestimated	F_R from Dowell
Thickening without reordering	Resistance increases	Combine with interleaving
Fundamental only	Harmonic loss missed	Sum per harmonic
Ignoring inter-winding capacitance	More common-mode noise	Evaluate capacitance
Ignoring loss near gaps	Local hot spots	Check with FE

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