High-Frequency Transformer Design
Theory: frequency, flux density and cross-section
Overview
A switch-mode power supply transformer fits in your palm, yet a mains-frequency transformer of the same power is far heavier. What's the difference?
Frequency. Voltage on a winding makes the core flux change over time; frequency is how many times per second it reverses. The higher the frequency, the less flux is needed to support the same voltage, so the core can be slimmer and the turns fewer. But higher frequency increases core and winding losses, so the size can't shrink forever. Striking that balance is what high-frequency transformer design is about.
Choosing the number of turns
Relation between square-wave voltage $V$, turns $N$, frequency $f$, peak flux density $B_{pk}$ and core cross-section $A_e$. $N A_e$ is inversely proportional to $f B_{pk}$. For sine waves the factor 4 becomes 4.44.
The core-loss limit
$P_v$ is core loss per unit volume and $k$, $\alpha$, $\beta$ are material constants (the Steinmetz equation). For ferrites $\alpha$ is often about 1–2 and $\beta$ about 2–3. Since loss rises with frequency, flux density must fall to stay within the allowable temperature rise.
Skin effect in windings
At high frequency current concentrates within about $\delta$ of the conductor surface. Wire much thicker than $2\delta$ carries no current in its core, raising resistance. Proximity effect from neighbouring conductors adds to this, multiplying losses in multilayer windings.
Ferrite made the switching supply possible
Iron and silicon-steel cores conduct electricity, so at higher frequency eddy currents flow inside them and generate heat. In the 1930s Yogoro Kato and Takeshi Takei at the Tokyo Institute of Technology invented ferrite, an iron-oxide magnetic material that barely conducts. With almost no eddy currents, it made cores usable at tens of kilohertz and beyond. Combined later with transistors switching current rapidly, it gave rise to the small, light switching power supply now found even in smartphone chargers.
Worked examples
Example 1: 50 Hz versus 100 kHz
Supporting a 400 V square wave, with 50 Hz silicon steel at 1.5 T peak and 100 kHz ferrite at 0.15 T peak:
| Case | Turns × area | Turns with 2 cm² core |
|---|---|---|
| 50 Hz, silicon steel 1.5 T | 1.33 m² | about 6,700 |
| 100 kHz, ferrite 0.15 T | 0.0067 m² | about 34 |
Even with a tenth of the flux density, 2,000 times the frequency cuts the required turns × area to 1/200. Real 50 Hz transformers use much larger cores to bring turns down to a few hundred, but either way they need vastly more iron and copper.
Example 2: shrinking limited by core loss
For an illustrative ferrite (α = 1.4, β = 2.5, 100 kW/m³ at 100 kHz and 0.1 T), the flux density allowed at a core-loss limit of 200 kW/m³, and required turns × area relative to 50 kHz:
| Frequency | Allowed flux density | Turns × area | Copper skin depth |
|---|---|---|---|
| 50 kHz | 0.195 T | 1.00 | 0.30 mm |
| 100 kHz | 0.132 T | 0.74 | 0.21 mm |
| 200 kHz | 0.090 T | 0.54 | 0.15 mm |
| 500 kHz | 0.054 T | 0.36 | 0.093 mm |
| 1 MHz | 0.036 T | 0.27 | 0.066 mm |
Twenty times the frequency, but only about a quarter the size.
Right. Each frequency increase forces lower flux density to hold core loss, cancelling most of the gain. Skin depth also shrinks, so at 1 MHz strands thinner than 0.13 mm are needed. With rising winding loss and cost, products usually settle between 50 and 500 kHz. GaN and SiC transistors now keep switching losses low at higher frequencies, and designs using MHz-range materials and planar windings are growing.
Example 3: same transformer, different frequency
Take a transformer designed for 100 kHz and 0.15 T and run it at 50 kHz with the same voltage: flux density rises to 0.30 T and core loss about doubles (276 → 591 kW/m³). Many ferrites saturate near 0.4 T at high temperature, and approaching saturation can make current surge and destroy the circuit. Any change that lowers frequency must be checked for flux density and temperature.
Analysis and design practice
Procedure
- From frequency and allowable temperature rise, set the core-loss limit and usable flux density.
- Set turns and core size from Faraday's law.
- Check skin depth, choose litz, foil or planar windings, and estimate winding loss.
- Use magnetic FEA for leakage inductance and winding-loss distribution.
- Check peak temperature with thermal analysis and revise flux density if needed.
“Core loss matches, yet it runs hot”
A prototype transformer's core loss matches the datasheet calculation, yet it runs 20–30 K hotter than expected. Usually the culprit is the winding: copper loss computed from DC resistance ignores skin and proximity effects, and in multilayer windings AC resistance can be several times DC. Near core gaps, fringing flux also penetrates the winding and drives local eddy currents. Magnetic analysis including winding arrangement and gap position is the reliable way to confirm the loss distribution.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Assuming size shrinks in proportion to frequency | Overheating from core loss | Set flux density from the loss limit |
| Copper loss from DC resistance | Winding loss underestimated | Include skin and proximity effects |
| Not checking saturation when hot | Current surge and failure | Check with saturation flux at max temperature |
| Windings near the gap | Local eddy-current heating | Keep away or distribute gaps |
| Casually lowering frequency | Flux density rises to saturation | Recompute turns and flux density |
I'd like to try transformer design myself.
Try the transformer design tool for turns and core, the flyback converter tool for flyback transformers, the skin depth calculator for skin depth and the transformer efficiency tool for efficiency. For detailed loss estimation see the Steinmetz equation, winding loss, leakage inductance and planar transformers.
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