Transformer core window area vs core area: selecting the right proportions

1. Quick alignment: what we mean by Aw, Ac, Kw, Ku

No long theory here, just labels so we talk about the same things.

• Ac – effective magnetic cross-section of the laminated stack, after stacking factor (net ≈ gross × stacking factor; ~0.95 for typical Si-steel laminations, ~0.8 around there for amorphous metal).
• Aw – usable window area for copper + insulation in one window.
• Ap – area product, Ap = Aw · Ac, used by most core makers to rate power capability.
• Kw – window space factor, copper area / total window area.
• Ku – window utilization factor (how much of the window is actually copper once you account for insulation system and winding style).

Power-level formulas can always be pushed back to something of the form

Aw · Ac ∝ P / (Kw · Bmax · J · f)

for a chosen flux density, current density, frequency, and utilization.

So Ap gets you how big the core has to be. This article is about how to split that Ap between Aw and Ac when you order or design the lamination stack.

2. Why the Aw / Ac proportion matters even if Ap is “right”

Take two lamination stacks with the same Ap:

• Core A: big Ac, tight window
• Core B: smaller Ac, generous window

Both satisfy the power equation on paper. They won’t behave the same once you wind, insulate, and ship them.

A few examples you probably recognise:

1. Magnetizing current vs copper losses
   • Push Ac up (for the same flux): magnetizing current drops, core loss per kg improves.
   • But your window shrinks, so copper runs hotter or you drop a wire gauge.
2. High-voltage insulation
   • Same Ap, but if most of it is Ac, there is no room left for barriers, duct spacers, axial clearances.
   • Kw and Ku crash long before thermal design is happy.
3. Mechanical stack and cost
   • Bigger Ac at same Ap often means thicker stack or wider tongue. Both want more steel, more punching tonnage, more annealing load.
   • Bigger Aw at same Ap wants taller yokes, longer strips, more scrap.

So when buyers send us drawings that just say “Ap ≥ X cm⁴”, it is half the story. The ratio sets how stressful life will be for copper, insulation, and lamination tooling.

3. A practical way to think about Aw vs Ac

A rough mental model that works quite well in lamination-stack discussions:

• Ac is for the magnetic limit (Bmax, core loss, magnetizing current).
• Aw is for the thermal and insulation limit (J, winding temperature, clearances, creepage path).

Once Ap is fixed by the power requirement, you slide along a line of solutions:

• Slide towards larger Ac, smaller Aw → magnetics get easier, winding and insulation get harder.
• Slide towards larger Aw, smaller Ac → winding and insulation relax, magnetics and no-load loss get harder.

That’s all. The rest is you deciding which side of the problem you’d rather fight on.

4. Typical design “zones” for laminated stacks

Instead of absolute numbers (which bounce around with material and frequency), it’s usually more useful to talk about bias — designs that clearly favour Ac or Aw.

Below is a qualitative table you can slide into your spec discussions.

Table: how different Aw / Ac biases show up in real transformers

You can read the table row that matches your world and already know where you should probably nudge the ratio.

5. How this ties back to lamination stacks in practice

From the lamination side, tweaking Aw and Ac usually comes down to a few levers.

5.1 Changing Ac without breaking the window

For laminated EI / UI stacks:

• Increase stack height Simple, but only up to what your yoke clamp and tank height allow.
• Change lamination shape (e.g. from 2-step to 3-step cruciform) This improves core area factor (net Ac vs circumscribing circle) and makes better use of tongue width.
• Use better material or higher Bmax Sometimes you accept a bit more core loss instead of increasing Ac.

All three give magnetic headroom while keeping Aw almost the same.

5.2 Changing Aw without upsetting Ac too much

• Taller window Increase window height while adjusting stack or yoke dimension so that Ac stays inside target.
• Wider window / bigger leg spacing (core-type) Legs move apart, window width grows; you might need to adjust tongue width to recover Ac.
• Different core family (shell vs core) Shell-type cores buy more window per unit of Ac, at the cost of more complex winding.

On a lamination drawing this often shows up as new values for:

• window height / width
• limb spacing
• tongue width
• stack thickness

So when you negotiate a change with your lamination supplier, it’s useful to say explicitly which of these you’re willing to move, not just “I need 10% more Aw”.

6. Quick rule-set for selecting Aw / Ac proportions

This is the short, slightly rough guide we see work well in B2B projects.

Step 1 – Start from losses, not from catalogue Ap

1. Pick your target:
   • allowed no-load loss
   • allowed load loss
   • top-oil / winding temperature
2. From that, pick Bmax and J that your organisation is comfortable with.

Then compute the minimum Ap from your usual formula set or from the high-frequency or power-frequency guide you use.

Only then open lamination tables.

Step 2 – Choose a “bias” based on the project

• Short-duty, low-voltage, high-current (welders, power tools, chargers) → okay to let Aw be tighter and push Ac up, because copper is chunky and runs cool between cycles.
• Continuous-duty distribution → stay nearer to balanced Aw / Ac. Neither copper nor steel gets a free pass.
• High-voltage, many windings, insulation-heavy → accept window-heavy layouts. You’ll pay in core kilos, but you avoid re-winding in production.

Write this bias into the spec. One line is enough:

“Prefer core-heavy / balanced / window-heavy solution for given Ap.”

It saves a lot of back-and-forth.

Step 3 – Check window space factor the way your winding shop actually winds

Once a tentative Aw is known:

1. Compute Kw from your copper areas and window area.
2. Back out Ku using your typical insulation system and winding style (layer vs random).

Then sanity-check:

• If Kw is under, say, 0.25 for low-voltage units, you are probably wasting window.
• If Kw is above ~0.45-0.5 in a high-voltage design, you’ll struggle with real clearances, even if it fits in CAD.

Adjust Aw or number of windows before changing anything in Ac.

Step 4 – Re-check Ac with real stacking factor and tolerances

Design calculations often assume a neat stacking factor. Real lamination stacks do not.

For each candidate lamination set:

1. Apply the realistic stacking factor from your supplier test data (varies with thickness and coating).
2. Subtract expected burr, deburr, varnish, and clamping variation.
3. Re-evaluate Bmax at rated flux.

If your “window-heavy” choice pushes Bmax too close to knee, either:

• go up one core size, or
• re-bias towards more Ac, or
• revisit the allowed flux density.

Doing this before the purchase order is a lot cheaper than asking for more stack height after tooling is made.

7. Special cases: high-frequency and matrix transformers

For high-frequency laminated or ferrite designs, the same Aw · Ac logic still holds, but the trade-offs shift.

Papers on solid-state transformers and matrix transformers show that at fixed Bmax and current density, transformer volume scales roughly with Ap / f, with volume minima when Aw and Ac grow in a certain proportion that balances copper and core volume.

In practice:

• Going to very high frequency often makes window the dominating constraint. Skin effect, proximity, and clearances all want extra room.
• The lamination stack itself may change to tape-wound or cut cores, which ties Aw and Ac geometry together more tightly than EI stacks.

So even though formulas become more involved, the old question is still there:

“Given the Ap, do you want a bigger window and smaller core, or the other way round?”

You gain a lot by making that choice explicit early.

8. What to actually put on a lamination drawing

If you want your lamination supplier (or internal stamping shop) to hit the Aw / Ac compromise you had in mind, giving just Ap and stack height is not enough.

Include, at minimum:

• Target Ap and core type (EI, UI, shell, step-lap, wound).
• Target Aw and window aspect ratio range (e.g. 2.5–3.5).
• Target Ac with allowed ±% based on stacking factor tests.
• Allowed band for Kw at rated voltage (derived from your insulation system).
• Which side you are okay to change if there is a conflict:
  • “Increase stack but keep window”
  • or “Open window more, accept higher Bmax”

Even these few extra lines stop your lamination stack from drifting back to the generic balanced point that may not fit your actual winding and insulation.

9. FAQ