Toroid Calculator

What Is the Toroid Calculator?

How to use the toroid calculator

1. Choose the unknown. The Solve for menu opens on Magnetic field, and the other three choices are Number of turns, Winding current and Distance from the ring axis.
2. Type the number of turns. A plain count from 1 to 100000, opening on 500, with presets at 10, 500 and 2000. Wire gauge and winding pitch do not enter the arithmetic at all.
3. Type the winding current. It takes A or mA and opens on 2 A, with presets at 0.05, 2 and 10 A. The box accepts 1e-6 to 1000 A.
4. Type the distance from the ring axis. It takes cm, mm or m and opens on 4 cm. This is measured from the centre of the hole — not the ring's own radius and not the gap to the nearest wire.
5. Read the answer. The headline carries the answer to six significant figures, trailing zeros dropped, in the unit that box was showing, and it switches to scientific notation below 0.0001 and at or above 10 million. Set the unit first, point Solve for at the target second and type your figures last — switching the target wipes every box back to its default and only the unit menus survive it.
6. Read the gauss reading. The same field in the unit most magnet datasheets still use. One tesla is exactly ten thousand gauss, so it is a change of unit rather than of quantity.
7. Read the turns per metre. How closely the winding is spaced along the circle you asked about, which is the one number that connects a ring to a straight coil.
8. Read the field at twice the radius. Always exactly half the answer, because the field goes as one over the distance. It is the real field out there only if the core reaches that far.
9. Open Show working. The steps give the relation, the boxes in SI base units, the substitution, what the distance is measured from, the ampere-turns, the gauss conversion, the turns per metre, the cost of doubling the distance, the assumptions and every unit conversion.
10. Try the inverse modes. Solve for the turns to size a winding, for the current to see whether a coil is buildable, or for the distance to find where a field has fallen to a given value.

This page exists for the one step that turns a winding into a field. The toroid guide is the prose half of it, where the argument about the field outside the ring is made and the problems are worked. What the current in the winding actually is, as charge per second, belongs to the electric current guide instead.

Two mistakes account for most wrong answers here. The first is entering the ring's own radius, or the distance to the nearest wire, where the distance from the axis belongs. The second is expecting the cross-section to matter: the shape of the core decides which radii exist, not what the field is at a radius that does exist.

Worked example: change one thing at a time

The table starts at the defaults, moves the unknown through all four modes, then loads the real cases, shows three different windings giving one field, walks every unit menu, and finally reaches the entries the calculator declines. Every answer and reading cell was taken from the running widget rather than worked out by hand. Where a cell and the tool ever part company, believe the tool.

Rows 1 to 4 are one state — 500 turns at 2 A read 4 cm from the axis — with the Solve for menu in each of its four positions. All three reading columns are identical down all four rows, because the readings describe the state rather than the mode. That is not free: they are worked out from the answer substituted back, so the mode in which the turns are the unknown still has a turns count to take a turns-per-metre figure from.

Rows 5 and 6 are the two edges of the same core, and their ratio is the cleanest check on the page. 6.66667 mT divided by 4 mT is 5 over 3, which is exactly the outer radius divided by the inner one — and it stays 5 over 3 whatever you do to the turns and the current, because the non-uniformity is pure geometry.

Rows 9 to 11 are the point of the whole page. Five hundred turns at 2 A, 2500 at 0.4 A, 200 at 5 A and 1000 at 1 A all read 5 mT at the same radius, because all four are a thousand ampere-turns. Only the turns-per-metre column moves, because that one does depend on how many turns there actually are.

Rows 12 and 13 are the two scalings. Doubling the distance from 4 cm to 8 cm halves the field from 5 mT to 2.5 mT, which is the reading row 1 already printed in its third column. Dropping from 500 turns to a single turn divides the field by 500, giving 0.01 mT.

Rows 15 to 18 run the three inverse modes, and row 17 is the one worth trying. Ask where the field has fallen to 4 mT on the opening winding and the answer is 5 cm, which is the outer edge of the reference core — the same pair of numbers as row 6, read from the other end. Rows 19 to 24 change nothing but the units: 50 G, 13.333333 uT, 50 mA, 40 mm and 0.04 m all reproduce the figures they came from, and the field box set to tesla prints the same answer as 0.005 T.

Rows 26 to 31 are the refusals: zero turns, a negative current, a distance of zero, more turns than the box accepts, a field above 10 T, and a field so weak that the distance it implies is no longer a ring. In every one of them the answer is words rather than an infinity or a NaN. If what you wanted was the flux through a loop in a field that is already uniform, the magnetic flux calculator is the tool for that instead.

Formula and symbol reference

One relation does all the work: B = mu0 · N · I / (2 · pi · r), for any distance r from the axis that lies between the inner and outer edges of the core. Everything else on this page is that relation rearranged, or a quantity derived from the state it describes.

The combination that makes it a one-line calculation is mu0 / (2 · pi) = 2.0000000011 × 10-7 T·m/A, so the field is that constant times the ampere-turns over the distance. Reading it backwards gives N = B · r · 2 · pi / (mu0 · I), I = B · r · 2 · pi / (mu0 · N) and r = mu0 · N · I / (2 · pi · B). The straight-coil form of the same question, which asks for turns per metre instead of a distance from an axis, is the magnetic field calculator.

There is deliberately no box for the core cross-section, the ring height or the permeability. The first two decide which radii exist rather than what the field is at a radius that does exist, and the third is fixed at 1 because everything here is air-cored — a ferrite or iron core multiplies every field on this page by its own relative permeability. Where cores and coils are the subject, that is the electromagnet guide.

The physics: why three numbers are enough

Draw a circle round the axis of the ring, inside the core. The winding is idealised as a smooth current sheet with perfect axial symmetry, so the field has the same magnitude everywhere on that circle and runs along it. That is what makes the loop integral of the field simply the field times the circumference.

What the circle encloses is the wire, N times over, each turn carrying I. So the enclosed current is N · I and nothing else, which is why the calculator needs no geometry: the circumference supplies the 2 · pi · r and the winding supplies the ampere-turns. Everything the field depends on is in those two numbers.

That also fixes the one-over-distance behaviour exactly. The enclosed current does not change as the circle grows, but the circumference does, so the field has to fall in step with it. Doubling the distance from the axis halves the field, which is the third reading beside every answer.

The same fact read the other way gives the bridge to a straight coil. The winding has N / (2 · pi · r) turns per metre along that circle, and a straight coil of that spacing carrying the same current has exactly this field — so a ring fat enough that the distance barely changes across it really is a long coil bent round. What a field like this then does to a charge moving through it is a separate question, and it belongs to the magnetic field guide.

Outside the ring the ideal answer is exactly zero, because a circle drawn in the hole or beyond the outer edge encloses no current at all. That is a property of the idealised winding and not a measurement of a real one, which is why this calculator asks for a distance it assumes is inside the core and says so rather than drawing a boundary it cannot know about.

Where the ideal toroid breaks down

The arithmetic here is exact to the last digit a double can hold. What goes wrong is physical: a winding treated as a smooth, perfectly symmetric sheet, a core with no material in it, and a single distance standing in for a whole cross-section.

A real winding is not axially symmetric close in

Looking out from the axis, a real winding is sometimes on the inside of the core and sometimes on the outside, so the field near the wires is not the smooth circle this relation assumes. The source the cluster works from puts the recovery distance at several times the winding spacing: further out than that, a real toroid does look symmetric. Nothing on this page corrects for it, and no figure for the size of the correction is printed.

A real winding always carries one loop round the ring

However many times the wire encircles the core and however thin it is, the winding also goes once round the plane of the ring, and that loop has a field of its own. This is the second condition the ideal result needs and the second one a real ring breaks. The honest claim is that leakage can be made low, not that it is absent, and the practical consequence is less interference rather than none.

Zero outside belongs to the ideal case only

The exactly zero field outside the ring is a property of the model. It follows from the two assumptions above, so a sentence that uses the word exactly has to say which case it is talking about. This calculator never asks about a point outside the core, which is why it has no boundary to report.

One distance is not a cross-section

The field is different at every radius across the core, so quoting a single figure for the field in a ring is already an approximation. The mid-radius value understates the average across a rectangular cross-section by a little over 2 per cent on the cluster reference core, and by more than 15 per cent on a ring whose outer radius is four times its inner one. This page prints the field at the radius you asked for and makes no claim about the average.

Air-cored throughout

Relative permeability is 1 everywhere on this page, so a ferrite or laminated-iron ring is a different number: every field here multiplies by the core's own relative permeability. No value for that is published or assumed anywhere on this page, and applying it is a separate step.

Steady currents only

Everything here is magnetostatics: one current, not changing. A current that varies in time brings in effects this relation knows nothing about, and none of them are modelled, estimated or hinted at by any reading on the page.

The turns are a whole number, and the answer may not be

Solving for the turns returns whatever the field and the current ask for, which is usually not an integer. The working prints the rounded figure beside the unrounded one and works out what rounding costs for your own answer, in per cent. On any winding of more than a few hundred turns that cost is far smaller than the tolerance of a ring you could actually build.

A clean answer can be a coincidence of the numbers

The opening state reads 5 mT exactly, and that is because mu0 / (2 · pi) happens to be 2.0 times ten to the minus seven to ten figures and 500 times 2 divided by 0.04 happens to be 25000. It is an accident of the chosen figures and not a law. Change any box and the roundness disappears.

The field box is range-checked, not merely checked for sign

Solving for the distance from a field of 1e-300 tesla returns a perfectly finite answer of about 2e+296 metres, which passes any test for being greater than zero. That is why every box here is checked against its declared range, and why the page declines rather than printing a figure that is arithmetic rather than physics.

The calculator measures nothing — you supply the numbers

Three quantities go in, one relation is used, and the answer describes the ring you described. The case names are shorthand for the four figures behind them and are not claims about any particular component, core or sample.

Where this field is actually used

Sizing a winding for a field you need

Point Solve for at the turns and the question becomes the one a build actually asks: you know the field you want, the current you can supply and the radius you have, and you want the turns. At 10 mT on 1 A at 2 cm the answer is 1000 turns, and the working prints the rounded figure beside it because a winding is a whole number.

Deciding between many turns and a heavy current

Because only the ampere-turns matter, the two are interchangeable, and the current mode is where you find out which is cheaper. 20 mT at 2.5 cm needs 10 A through 250 turns, or a tenth of that through ten times as many. The turns-per-metre reading is what tells you whether the winding will physically fit.

Reading a field measurement back into a distance

Point Solve for at the distance and a probe reading becomes a position. On the opening winding a reading of 4 mT places you at 5 cm from the axis, and 5 mT places you at 4 cm. That is a useful consistency check on where a probe actually sat.

Judging how uniform a core is before trusting one figure

Enter the inner radius, then the outer radius, with the turns and current unchanged. The reference core gives 6.66667 mT and 4 mT, a spread of two thirds, and a ring with four times the radius ratio is far worse. A narrow annulus is nearly uniform and a wide one is not, which is the arithmetic behind making a core fat rather than wide.

Working in gauss when the datasheet does

Core and magnet datasheets are still quoted in gauss far more often than in tesla, and the first reading beside the answer saves the conversion. 5 mT is 50 G and 61.5385 mT is 615.385 G. The step in the working says the conversion is exact, so nothing is lost by moving between them.

Checking a ferrite bead rather than a transformer

Load a ferrite bead on a signal lead and the same relation drops 375-fold, to 0.0133333 mT on ten turns at 50 mA — the headline stays in mT because that is the unit the field box is set to, and step 3 of the working prints the identical field as 13.333 uT. The field a small suppression component produces is tiny, and seeing that in the same units as a mains core is the quickest way to get the scale right. The real bead has a ferrite core, so its field is larger by that core relative permeability.

Recognising the same relation somewhere else

One over the distance from a line, times a current, is the field of a long straight wire — and the ring is exactly that law with N wires worth of linked current instead of one. Once you see that, the two formulas stop being two things to remember. Where this field appears as flux through something, the magnetic flux calculator takes it from there.

Where to go next

For the argument this arithmetic supports, the three tests of the field inside a ring and the problems worked through, read the toroid guide, which this tool is the calculating half of. The solenoid guide is the straight-coil case this one is the bent version of, and the magnetic field guide is what a field like this does to a moving charge.

Four more posts pick up the pieces this page uses but does not explain. The electric current guide is what the current in the winding is, the electromagnet guide is where cores and coils are the subject, the magnetic flux guide is the flux this field carries through a surface, and the right-hand rule guide is which way round the ring it runs.

Three tools sit next to this one. The magnetic field calculator is the straight-coil form, the magnetic flux calculator takes a uniform field through a loop, and the electric current calculator gives the current from charge and time. The solenoid simulator and the electromagnet simulator are the interactive versions of the first two cases.

The full physics lab library and the calculator index are open too.