Eddy Current Loss Calculator

What Is the Eddy Current Loss Calculator?

How to use the eddy current loss calculator

1. Choose the unknown. The Solve for menu opens on Power loss per unit volume. The other four choices are Peak flux density, Frequency, Lamination thickness and Resistivity, and whichever you pick vanishes from the boxes below.
2. Type the peak flux density. Peak flux density takes T, mT or G and wants the peak of the alternating field rather than a root-mean-square value. It enters the relation squared, so −0.50 T returns exactly the same loss as 0.50 T; only a field of exactly zero is declined.
3. Type the frequency. Frequency takes Hz or kHz and must be above zero. It does two jobs at once: it raises the loss as its square, and it shrinks the skin depth as one over its square root, which is why a thickness that is safe at 50 Hz may not be at 400 Hz.
4. Type the lamination thickness. Lamination thickness takes mm, m, um or in, and it is the thickness of one sheet across the field, not the height of the stack. Forty half-millimetre sheets are 0.500 mm here, not 20.0 mm, and getting that wrong is the single commonest mistake on this page.
5. Type the resistivity. Resistivity takes uOhm.cm, Ohm.m, uOhm.m or nOhm.m and opens on copper at 1.68 uOhm.cm. It must be above zero, because a perfect conductor would divide the formula by zero and a negative resistivity would hand back a negative power.
6. Type the volume of metal. Volume of metal takes cm³, m³ or L, and it never moves the headline. It is there to turn a power density into watts in the Total loss in that volume chip, and it is the one field that can never be the unknown.
7. Read the answer and the chips. The headline carries six significant figures while it stays a plain decimal, and turns into a four-figure exponential once it passes ten million in the unit the box is set to — the solid-block case reads 2.448e+7 W/m³ until you switch the box to MW/m3, where it reads 24.4782. The chips give the total loss, the skin depth, the thickness in skin depths, the exact slab loss, the ratio between the two, a worded validity check, and the loss at half the thickness and at twice the frequency.
8. Read the validity chip before you quote anything. Textbook formula check compares the headline with the exact slab solution and says in words when the headline is reading high. On the opening case the chip prints reads 0.00 percent high - inside the 1 percent line at d over skin depth 0.7051; on the same copper in one 20.0 mm block it prints READS HIGH by 88.28 percent - past the 10 percent line at d over skin depth 1.2542; read the exact slab figure instead.
9. Open Show working. The steps restate the relation, list your figures in the units the arithmetic really uses, print the top and the bottom of the fraction separately and divide them. The three modes that take a square root state plainly that a negative root exists and that the positive one is returned.

Four calculators here already own the neighbouring questions, and none of them can answer this one. The Faraday’s law calculator gives the EMF a changing flux drives round a coil of known turns, but it has no thickness and no resistivity; the resistivity calculator is R = ρL/A, the geometry of a wire, and it is where the 1.68 uOhm.cm copper figure on this page comes from. The magnetic field calculator and the magnetic force calculator both describe a steady field, and nothing is induced by a field that does not change.

The six-line derivation, the drawn loops and eight worked problems are all next door in the guide to eddy currents, which this tool is the arithmetic half of. If the direction of the induced loops is the unfamiliar part rather than their size, the guide to Lenz’s law is the place to start, and the Lenz’s law simulator shows the same opposition happening in a coil.

Two mistakes account for most wrong answers here. The first is typing the thickness of the whole stack instead of one sheet, which multiplies the answer by the square of the number of laminations; the second is quoting the headline without reading the validity chip, which is how a solid block gets reported as losing 88.28 per cent more than it really does.

Worked example: change one thing at a time

The table starts at the defaults and moves one thing at a time: the thickness, then the frequency, then the metal, then which quantity is the unknown, then the unit the figures are typed in, and finally the entries the calculator declines. Rows 8 to 11 are the closed round trip, so the loss that leaves row 5 comes back in and returns the thickness, the frequency, the field and the resistivity it started from. Every Result, Total loss and Textbook / exact cell was read out of the running widget rather than worked out by hand, so where a cell and the tool ever part company, believe the tool.

Rows 1 and 2 are the whole point of lamination, and they are the same 100 cm³ of copper in the same field. The headline reads 15298.9 W/m³ for the 0.500 mm sheet and 24.4782 MW/m³ for the one 20.0 mm block — but the block sits at 2.168 skin depths, where the last column shows the formula running 88.28 per cent high. Its exact slab chip reads 1.300e+7 W/m³ instead, and that, not the headline, is the figure to compare the sheet against.

Rows 3 and 4 raise the frequency to 400 Hz, eight times the mains figure, and the thin sheet duly loses 64 times as much because the relation squares the frequency. The block does not follow. Its skin depth falls from 9.225 mm to 3.262 mm, so the same 20.0 mm is now 6.132 skin depths and the headline overstates the loss by a factor of 38.56.

Rows 5 to 7 change the metal. Row 5 posits a 50.0 uOhm.cm metal, nearly thirty times the resistivity of copper, and the thin-sheet relation divides the loss by that same factor because it goes as one over rho — the relation restated, not a finding. Nichrome at 110 uOhm.cm keeps even a solid 20.0 mm block inside the 1 per cent line, and row 6 halves row 5’s thickness and quarters its loss for the same kind of reason.

Rows 8 to 11 read the relation backwards in each of the four remaining modes. Asking for a loss of 10000 W/m³ gives a maximum thickness of 1.10266 mm, and the last three take the 2056.167584 W/m³ that row 5 produces and return 50 Hz, 1 T and 50 uOhm.cm exactly. Three of those four rearrangements have a negative root as well, which the calculator discards.

Rows 12 to 14 are the unit menus and the sign. 500 mT is the same field as 0.50 T and 500 um is the same sheet as 0.500 mm, so both reproduce row 1 to the last digit. So does a field of −0.50 T, because the relation squares it.

Rows 15 to 17 sit on the edge of the domain and are refused there. A frequency of zero, a field of zero and a negative resistivity are all declined rather than answered, because the bare formula returns exactly 0 W/m³ for the first two and a negative power for the third, and a plausible-looking number is far more dangerous than a refusal.

Formula and symbol reference

The calculator uses one relation in five arrangements, and carries a second one beside it as a check. Apply Faraday’s law to a rectangular loop inside a slab of non-magnetic metal, average the dissipation across the thickness and over a cycle, and the loss per unit volume comes out as P/V = π2B2f2d2 / (6ρ). There is no fitted constant in that: the 6 is 12 times 2, the 12 from the second moment of the thickness and the 2 from the mean square of a cosine.

The second relation is the skin depth, δ = sqrt(ρ / (πfμ0)), and it is the yardstick rather than a decoration. The derivation above quietly assumed that the field is the same all the way through the sheet, which is true only while the thickness is small compared with δ. Solving the same slab exactly, with no such assumption, gives P/V = B2(sinh s − sin s) / (σμ2dδ(cosh s + cos s)) with s = d/δ, and that is the figure in the exact slab chip.

Each rearrangement below is the exact form one of the Solve for modes evaluates, which is why they are set out rather than left implied. Reading the relation backwards gives B = sqrt(6ρ(P/V) / (π2f2d2)), f = sqrt(6ρ(P/V) / (π2B2d2)) and d = sqrt(6ρ(P/V) / (π2B2f2)), each with a second, negative root that the calculator discards; solving for the resistivity is linear, ρ = π2B2f2d2 / (6(P/V)), and has one root only.

The physics: why thin sheets win, and where the formula stops telling the truth

Put a lump of metal in an alternating field and Faraday’s law does not care that there is no wire. Every closed path inside the metal encloses a changing flux, so an EMF appears round every one of them, and the metal’s own conductivity turns that EMF into current. The guide to electromagnetic induction covers the EMF itself; what this page adds is what happens once that EMF has nothing but metal to drive current through.

The direction is settled by Lenz’s law: the loops flow the way that opposes the change that made them. That single sentence is the whole of the qualitative story and it belongs to the Lenz’s law article rather than to this page. What Lenz’s law cannot tell you is how much power the loops burn, because it contains neither a thickness nor a resistivity.

The size comes from the geometry. A loop a distance x from the mid-plane of the sheet encloses a flux proportional to x, so the induced electric field grows linearly outwards and the dissipation, which goes as the square of it, grows as x squared. Average that across the thickness and the answer picks up a factor of d2/12, which is where the square comes from.

That square is why lamination works so well, and why the benefit is easy to overstate. Slice a block into ten insulated sheets and each sheet is a tenth as thick, so each loses one hundredth as much per unit volume while the volume of metal is unchanged — a hundredfold cut in total. The insulating varnish does no physics of its own; it simply stops the loops from closing across the whole block.

Now the part that a page written from memory usually gets wrong. The derivation assumed that the field is the same everywhere across the thickness, and the eddy currents themselves make that false: their own field opposes the applied one, so beyond about a skin depth the interior of the metal is screened and carries almost nothing. The thin-sheet formula keeps growing as d2 for ever; the metal does not.

The boundaries were measured rather than guessed, and they are the most useful numbers on this page. The textbook formula is good to 0.5 per cent while the thickness stays under 0.5929 skin depths, to 1 per cent under 0.7051, to 5 per cent under 1.0545, and by 1.2542 it is 10 per cent high. Real laminations sit far inside all of that; a solid block usually does not.

Past those lines the disagreement stops being academic. The same 100 cm³ of copper in one 20.0 mm block sits at 2.168 skin depths, where the headline reads 88.28 per cent high, and at 400 Hz the same block sits at 6.132 skin depths and the headline is out by a factor of 38.56. The exact slab chip is unaffected by any of this, because it makes no thin-sheet assumption at all.

There is a genuine surprise hiding in that. Because only a skin of order δ carries any current, the loss per unit volume of a thick slab peaks at 2.2542 skin depths and then falls: by ten skin depths it is down to 23.97 per cent of its peak. Adding metal below the skin adds volume without adding loss, which is the exact opposite of what d2 promises.

It also means the familiar scaling rules quietly stop applying once a sheet is thick. Doubling the frequency at two skin depths multiplies the exact loss by 1.8762 rather than by 4; doubling the thickness there multiplies it by 0.6478, so the loss goes down; and doubling the resistivity multiplies it by 0.7067 rather than halving it. The two what-if chips on this page are the thin-sheet rule restated and carry none of that, which is why the validity chip sits between them and the headline.

Finally, a limit worth naming before it bites. Everything above is about a non-magnetic metal, and a real transformer core is not one — it is a ferromagnetic alloy whose relative permeability runs into the thousands, which shrinks its skin depth by the square root of that same factor. That is why real laminations are a fraction of a millimetre thick, and it is also why no figure on this page is offered for any steel.

Where the eddy current loss calculator breaks down

The arithmetic is exact to the last digit a double can hold. What fails is a thin-sheet formula being asked about a thick sheet, a ferromagnetic core being described by a non-magnetic model, or a stack thickness being typed where one lamination belongs.

The headline formula is an approximation, and its boundary is printed for you

Everything in the banner rests on the field reaching right through the sheet, which stops being true once the thickness approaches the skin depth. Measured against the exact slab solution, the formula is 1 per cent high at 0.7051 skin depths, 5 per cent high at 1.0545 and 10 per cent high at 1.2542, and it keeps going: at 2.168 skin depths it is 88.28 per cent high and at 6.132 it is out by a factor of 38.56. That is why the Textbook formula check chip exists and why it is worded rather than numeric.

The metal is non-magnetic, with a relative permeability of one

The skin depth used here is sqrt(ρ / (πfμ0)), with no relative permeability in it. A ferromagnetic core has a relative permeability in the thousands, so its skin depth is smaller by the square root of that factor and a sheet that looks thin in millimetres may be several skin depths thick. This page gives no figure for electrical steel or any other magnetic alloy, because none was measured for it; treat any such number you find elsewhere as needing its own source.

Hysteresis is a separate loss and is not modelled at all

A real ferromagnetic core loses energy twice over: once to eddy currents, which is what this page computes, and once to hysteresis, the work of turning the magnetisation round and back each cycle. Hysteresis rises roughly in proportion to the frequency while eddy-current loss rises as its square, so which one dominates depends on where you are. A core loss quoted by a manufacturer normally includes both, and comparing it with this page’s figure alone will not balance.

The field is assumed uniform, sinusoidal and along the sheet

One frequency, one amplitude, one direction, lying in the plane of the lamination. A square wave, a field with harmonics on it, a field that varies along the sheet or one with a component through the thickness all break the derivation, and a pulse-width-modulated drive is none of these things. The peak flux density box wants a peak rather than a root-mean-square figure, which for a sine wave differs by a factor of about 1.414 and therefore by a factor of two in the answer.

The laminations are perfect, and real ones are not

The model treats each sheet as perfectly insulated from its neighbours, infinitely wide and long compared with its thickness, and takes up no room for the insulation itself. A real stack has a stacking factor below one, burrs at the cut edges that can short neighbouring sheets together, and edges where the current has to turn round. All three push a real core above the figure here, and the edge effect matters most for a narrow sheet.

Type one lamination, not the whole stack

The thickness box is the thickness of a single sheet measured across the field. Typing the height of a forty-sheet stack instead of one sheet multiplies the answer by 1600, which is the square of the count, so it does not look like a slip — it looks like a catastrophe in the design. The volume box is where the size of the stack belongs, and it scales the total loss without touching the headline density.

Three of the five modes have a second, negative root

Solving for the field, the frequency or the thickness means taking a square root, and the relation squares all three, so the negative root satisfies it exactly as well as the positive one. The calculator returns the positive root and the working says so on its own line. Solving for the resistivity is linear and has a single root, and solving forwards for the loss has no ambiguity at all.

Zero is refused wherever it would look like an answer

At zero frequency, zero thickness or zero field the bare formula returns exactly 0 W/m³, which on screen is indistinguishable from a broken tool, so all three are declined as inputs. A resistivity of zero divides by zero and a negative one returns a negative power, so the guard there is strictly greater than zero rather than merely not equal to it. The refusal sits at zero and not near it: a hundredth of a hertz is a legal frequency and answers normally.

Do not divide the two power figures on screen

The headline and the exact slab chip are each rounded on their own before they are printed, and the ratio between two rounded numbers is not the rounded ratio. Over a grid of 60,000 states spanning the field from 0.05 to 2.00 T, the frequency from 10 to 1000 Hz, five thicknesses and three metals, dividing the two printed strings fails to reproduce the printed ratio in 5,178 of them, 8.63 per cent. Read the Textbook / exact chip instead, which is computed from the unrounded values.

The same applies to the skin depth chip

The skin depth is printed to three decimal places in millimetres, and recomputing the thickness in skin depths from that printed figure disagrees with the printed d over skin depth chip in 9,520 of the same 60,000 states, 15.87 per cent. A worked instance: at 20 Hz in copper the chip reads 14.587 mm, from which a 1.00 mm sheet works out at 0.06855 skin depths, while the chip beside it reads 0.06856. Both chips are right; neither is a check on the other.

Rounded figures in, rounded figures out

The headline is rounded while the arithmetic behind it is not, so feeding a printed answer back in as an input moves the result slightly. The reverse presets here land exactly because they type the ten-figure value that Show working prints, not the rounded headline. That is also why the working prints every operand unrounded: each line can then be checked exactly as it stands.

The two what-if chips are the formula restated, not a result

Loss if the thickness is halved is a quarter of the total and Loss at twice the frequency is four times it, because the relation says d2 and f2. They are on the page because they are the two most useful things to know about a lamination, not because the tool discovered them, and they inherit the headline’s assumption: once a sheet is a skin depth or more thick, the real metal does not follow either rule.

The calculator measures nothing — you supply every figure

Five numbers go in, one relation is rearranged, and the answer describes that idealised sheet and nothing else. It is not a measurement of a transformer, a motor or an induction hob, and the preset names are shorthand for the figures beside them rather than claims about any apparatus. Verify anything you mean to rely on against your own data before you quote it.

Where eddy-current loss is actually used

Why transformer and motor cores are stacks of thin sheets

This is the calculation that decides a lamination thickness. The loss goes as the square of the sheet, so halving it quarters the waste heat, and the only limits on going thinner are the cost of the sheet, the insulation between sheets and the fraction of the core volume it eats. Run the opening case, then run it again at 20.0 mm, and the difference between a stack and a lump is the whole of it.

Why aircraft and high-frequency equipment laminate harder

Aircraft electrical systems run at 400 Hz because a higher frequency lets a transformer do the same job with less iron, and switching supplies run far above that. The eddy-current loss rises as the square of the frequency, so the same sheet at 400 Hz loses 64 times as much as at 50 Hz, and the skin depth falls as well. Above a few kilohertz laminations give way to ferrites and to powdered-iron cores, whose resistivity is orders of magnitude higher.

Induction hobs and induction heating, where the loss is the point

Everything on this page is a description of waste until you want the heat, and then the same arithmetic runs in reverse: a coil drives a field into a pan and the eddy currents in the pan do the cooking. The design goal flips from thin sheets to a thick, reasonably resistive base, and the frequency is chosen to put the skin depth where the heat is wanted. Industrial induction hardening and metal melting are the same idea at much larger power.

Eddy-current brakes and damping

Drag a conductor through a field and the induced loops oppose the motion, which gives a braking force with no contact and nothing to wear out. It is used on trains and on roller-coasters, and in miniature on the aluminium disc of an old electricity meter and the needle of a moving-coil instrument. A magnet dropped down a copper pipe falls slowly for exactly this reason, and this page deliberately puts no number on that particular demonstration.

Metal detectors and non-destructive testing

A coil driven with alternating current induces eddy currents in any conductor nearby, and those currents push a field back into the coil and change what it sees. That is a metal detector; the same principle at a fixed, well-controlled frequency is eddy-current testing, which finds cracks and corrosion in aircraft skins and heat-exchanger tubes without touching them. What is being read there is the change in the loss, which is why the frequency and the skin depth matter so much.

Counting the cost of the waste heat

A core loss is a continuous draw whenever the equipment is energised, whether or not it is doing useful work, which is why transformer standby losses are regulated. Take the Total loss in that volume chip for the whole core volume and the electricity cost calculator will turn those watts into a yearly bill. The heat also has to be carried away, so the same figure sizes the cooling.

Where to go next

For the derivation in full, with the loops drawn, the screening explained and eight worked problems, read why metal cores are cut into slices. For the law that fixes which way the loops turn, the guide to Lenz’s law is the companion piece, with the Lenz’s law simulator beside it.

Three more are worth a bookmark. The guide to electromagnetic induction sets out the EMF this page turns into watts; the Faraday’s law calculator handles that EMF in a coil of known turns; and the resistivity calculator is where the copper and nichrome figures used here come from. The whole physics lab library is open too, and the site search will find anything this page has not.