Magnetics: Inductor and Transformer Design

Because the gap is where the energy is stored, and an inductor whose job is to store energy is useless without one.

The physics. Energy density is \( \tfrac12 BH = \tfrac12 B^2/\mu \). At the same flux density, a material with \( \mu_r = 2000 \) stores two thousand times less energy per unit volume than air. So even a fraction of a millimetre of gap holds far more energy than the entire ferrite core around it.

Put numbers on it. ETD34, \( A_e = 97\ \text{mm}^2 \), \( \ell_c = 78 \) mm, \( \mu_r = 2000 \), 40 turns:

• Ungapped: \( L = 5.0 \) mH, but \( I_{sat} = B_{sat}\ell_c/(\mu_0\mu_rN) = 0.27 \) A. Stored energy \( \tfrac12LI_{sat}^2 = 184\ \mu\text{J} \).
• 1 mm gap: \( L = 195\ \mu\text{H} \) — 26 times less — but \( I_{sat} = 6.96 \) A. Stored energy 4.73 mJ.

26 times more energy, from a component with a twenty-sixth of the inductance. Inductance was never the figure of merit.

The factor is exactly the reluctance ratio \( \ell_g\mu_r/\ell_c = 25.6 \): gapping divides \( L \) by it and multiplies \( I_{sat} \) by it, so \( \tfrac12LI^2 \) rises by it. Equivalently, the gap now holds 96% of the stored energy and the ferrite holds 4%.

Two further benefits:

1. Predictability. Ferrite's permeability varies ±25% part to part and drifts strongly with temperature. Once the gap dominates the reluctance, \( L = \mu_0N^2A_e/\ell_g \) depends only on geometry — a large manufacturing gain.
2. Graceful saturation. A gapped core's inductance falls gradually as it approaches saturation rather than collapsing, which gives the control loop a chance to react.

And the case where a gap is wrong. A true transformer only transfers energy, so any it stores is magnetising current — pure overhead that must be supplied, reset and paid for in loss. A forward, push–pull or bridge transformer therefore wants the highest magnetising inductance available: no gap.

The rule: indirect converters store, so they gap. Direct converters transfer, so they do not.

Because core loss depends on the flux swing, not on the flux level — and the DC current, however large, produces no swing at all.

The mechanism. Hysteresis loss is the area of the B–H loop traversed each cycle; eddy-current loss depends on \( dB/dt \). A steady flux traverses no loop and has \( dB/dt = 0 \). Neither mechanism operates.

Only the ripple matters:

\[ \hat B_{ac} = \frac{\Delta I/2}{I_{pk}}B_{max} = \frac{1}{21}B_{max} \approx 0.048\,B_{max} \]

And the Steinmetz exponent makes that tiny. With \( \beta = 2.5 \):

\[ \frac{P_{core}}{P_{core,\text{full swing}}} = (0.048)^{2.5} = 5\times10^{-4} \]

Five hundredths of one percent of the loss the same core would suffer if the flux swung its full range.

The design consequence. A buck, boost or PFC inductor is almost entirely copper-loss dominated. So:

• Design to the saturation limit, not to a loss-limited flux density. Running at 0.3 T is fine.
• Spend the effort on the winding — strand size, layer count, interleaving — not on a lower-loss core material.
• Powdered iron and sendust cores, which have higher loss but higher \( B_{sat} \) and a distributed gap, are often better here than expensive ferrite.

The exact opposite applies to a transformer, whose flux swings the full range every cycle. Same material, same frequency, and completely inverted priorities — which is why one design rule cannot serve both.

One caution. The DC flux still counts towards saturation. The core must accommodate \( B_{dc} + \hat B_{ac} \), and it is the peak that saturates even though only the swing dissipates.

Because beyond about twice the skin depth, the extra copper carries no current but does provide more volume for eddy currents to circulate in.

Skin effect. AC current induces eddy currents within its own conductor, which cancel the current at the centre. The current density falls as \( e^{-x/\delta} \) from the surface, with

\[ \delta \approx \frac{66}{\sqrt f}\ \text{mm} \quad\text{— 0.21 mm at 100 kHz} \]

A conductor thicker than about \( 2\delta \) has a dead core carrying almost nothing.

Proximity effect, which is usually worse. The field from neighbouring turns also induces eddy currents, and in a multi-layer winding the field builds up layer by layer. The outermost layer of a ten-layer winding sits in ten times the field of the innermost, and the induced loss goes as the field squared.

Why thicker wire makes this worse. The eddy current loops driven by the external field are larger in a thicker conductor, so they dissipate more. Adding copper adds loss.

Concretely at 100 kHz:

• 1.0 mm wire, 4 layers: \( d/\delta = 4.8 \), \( F_R \approx 12 \).
• Same copper as 0.4 mm strands: \( d/\delta = 1.9 \), \( F_R \approx 2.5 \). Nearly five times less loss.

What to do, in order of value:

1. Interleave. Halving the peak MMF quarters the proximity loss, and it costs only winding time. Nothing else comes close for the effort.
2. More, thinner strands, each below \( 2\delta \).
3. Foil for high-current, low-turn windings — one turn per layer and excellent utilisation.
4. Litz wire where the frequency demands it — but genuinely transposed, since bunched wire gives far less than its strand count suggests.

And the trade to remember: interleaving increases interwinding capacitance, worsening common-mode EMI. A grounded screen between windings is the usual compromise.

Because it conducts, and eddy-current loss rises with the square of frequency.

The resistivity gap. Silicon steel is a metal at about \( 5\times10^{-7}\ \Omega\text{m} \). Ferrite is a ceramic at roughly \( 1\ \Omega\text{m} \) — two million times higher. A changing flux induces the same voltage in both, but ferrite's resistance is so high that almost no current flows.

Why lamination stops helping. Eddy loss in a lamination of thickness \( t \) goes as \( t^2f^2\hat B^2 \). To hold the loss constant while doubling the frequency, the thickness must be halved:

• 50 Hz: 0.35 mm — standard, easily stacked.
• 10 kHz: about 25 μm — possible, expensive, poor stacking factor.
• 100 kHz: about 2.5 μm — thinner than kitchen foil, impossible to handle, and the insulation between laminations would occupy more volume than the steel.

So the crossover is near 1 kHz. Below it, steel's high \( B_{sat} \) gives a smaller core. Above it, ferrite wins on loss despite needing four times the cross-section for the same flux.

The exception: amorphous and nanocrystalline alloys. Metals cooled from the melt fast enough never to crystallise, produced as 20–25 μm ribbon. They keep \( B_{sat} \approx 1.2 \) T while controlling eddy currents into the hundreds of kilohertz. Several times ferrite's price, and used where size or weight justifies it — large PFC chokes, common-mode chokes, aerospace magnetics.

And why ferrite's low \( B_{sat} \) matters less than it appears. Above 20 kHz a ferrite design is usually limited by loss, not saturation: a typical transformer runs at 100–200 mT against a \( B_{sat} \) of 350 mT. Raising \( B_{sat} \) would not help, because \( \hat B^{2.5} \) would make the losses intolerable long before saturation was reached.

The core is too small. Redistributing turns will not help, because the losses are already balanced.

Why balance means you are stuck. Total loss is roughly \( P = A/N^\beta + BN^2 \) — core loss falling with turns, copper loss rising with them. Differentiating shows the minimum lies where \( \beta P_{core} = 2P_{cu} \), so with \( \beta = 2.5 \) the optimum is \( P_{core} \approx 0.8P_{cu} \) — near equality.

At that point every trade is already exhausted:

• Add turns → core loss falls, copper loss rises more. Total increases.
• Remove turns → the opposite. Total increases.

The design is at its minimum-loss point, and the minimum is too high.

So the answer is more surface area. Options, in order:

1. A larger core. \( R_{th} \approx 53/(A_eA_w)^{0.54} \), so doubling the area product cuts \( R_{th} \) by 31% and reduces both losses, since there is more room for copper and more core volume for the same flux.
2. Raise the frequency. A transformer's area product goes as \( 1/f \), so doubling \( f \) halves the required core — but core loss rises as \( f^{1.3} \), so the net gain must be checked. Often worthwhile up to a point.
3. Improve cooling. Forced air can double the permissible current density, which is equivalent to a much larger core. Cheapest if a fan is present anyway.
4. Better core material. A lower-loss ferrite grade at the same size shifts the balance and permits fewer turns.

The diagnostic value of the split. Unequal losses tell you what to change without measuring anything else:

• Core ≫ copper: too much flux — add turns.
• Copper ≫ core: too little flux — remove turns, and check \( F_R \).
• Both high, roughly equal: the core is too small.

Which is why measuring the two separately — a thermocouple on the core and one on the winding — is the first thing to do with a hot magnetic component.

Because nothing in a magnetic component fails at a particular voltage or current — everything fails at a particular temperature.

What actually sets the limits:

• Wire enamel is rated by class — 130 °C, 155 °C, 180 °C, 200 °C. Exceed it and the insulation degrades, turns short, and the component fails.
• Bobbin plastic softens and deforms above its rating.
• Ferrite's Curie temperature — around 200–250 °C — is where it stops being magnetic entirely. But long before that, \( B_{sat} \) falls by about 30% from 25 °C to 100 °C.
• Safety approvals specify maximum temperatures for insulation systems, and exceeding them invalidates the certification.

So current and voltage are proxies, and poor ones. The same winding carrying the same RMS current dissipates quite different power at 10 kHz and 1 MHz, because \( F_R \) differs. The same core at the same flux density loses quite different power at different frequencies. Neither current nor voltage predicts temperature without the rest of the design.

What this means in practice:

1. The design converges on \( \Delta T \). Every other quantity — turns, wire, gap, core size — is adjusted until the temperature rise is acceptable, typically 40 K.
2. Cooling is a design variable. Forced air roughly doubles the permissible current density, which is worth as much as a core size increase.
3. Ambient temperature must be specified. A 40 K rise is fine in a 25 °C room and destructive in a 90 °C enclosure.
4. Measurement beats calculation. \( k_u \), \( F_R \), fringing loss and thermal path are all hard to predict. A thermocouple on the core and one on the winding, run to equilibrium at full load, settles the question in twenty minutes.

And it explains a catalogue oddity. The same inductor is sold with two ratings — a saturation current and an RMS current. The first is a magnetic limit; the second is thermal, and is usually quoted for a stated temperature rise. They are independent, and a design must satisfy both.