Find the inductance of an air-core coil, a toroid or any core with an AL value, or the turns to wind for a target, using Wheeler's formula instead of the ideal solenoid.
Inductance Calculator - Air Core, Toroid and AL Value Coils
Find the inductance of an air-core coil, a toroid or any core with an AL value, or the turns to wind for a target, using Wheeler's formula instead of the ideal solenoid.
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Coil inductance for three constructions on one page
Three kinds of coil, each with the formula that actually fits it: a single-layer air-core coil through Wheeler's formula, a toroid through its dimensions and the core's relative permeability, and any core whose maker prints an AL value. The calculator finds the inductance from the number of turns or the turns for a target inductance. 100 turns on a 20 mm form, 50 mm long, give 67.57 µH; the textbook long-solenoid formula would say 78.96 µH.
What the form asks, coil by coil
- Kind of coil - air-core coil (wire wound on a plastic form or on nothing), toroid with a known permeability, or a core with an AL value from its datasheet.
- What to find - the inductance from the turns, or the turns for a target inductance.
- Turns, or the target and its unit - turns may be fractional on an air coil; the target can be in nH, µH, mH or H.
- Size unit - millimeters or inches, for the air coil and the toroid.
- Dimensions - for an air coil its outside diameter and winding length; for a toroid the inner and outer diameter and the height.
- Relative permeability - toroid only. 1 for air or plastic, the datasheet's initial permeability for a magnetic core.
- AL value - AL cores only, in nH per turn squared, as printed by the core maker.
- Read the result - the inductance or the turns to wind, the equivalent AL, and a table of the same coil with other turn counts.
Why turns count twice and a core counts only when it is closed
Every turn adds to the magnetic flux, and the same flux then passes through every turn. That is the whole reason inductance grows with N²: 20 turns on a core with an AL of 250 nH give 100 µH, and 40 turns give 400 µH, not 200.
The formula most people learn first, L = µ0N²A ÷ l, assumes a coil so long that the field inside is uniform and the ends do not matter. Wikipedia's table of inductance formulas carries it with a correction factor, the Nagaoka coefficient, which is close to 1 only for a coil much longer than its diameter. For ordinary proportions the same table gives Wheeler's approximation, L = r²N² ÷ (23r + 25l) in microhenries with the radius and length in centimeters, valid when the length is at least a fifth of the diameter. A 10 mm by 200 mm coil lands within 1% of the ideal value (ratio 0.99); a 20 mm coil only 5 mm long reaches just 0.357 of it.
Magnetic cores are where most online coil calculators go wrong. Wikipedia's inductor article describes a coil on a straight rod: the field lines leaving one end have to return through the air to the other, which reduces the field, and a closed magnetic circuit, typically a toroid, is what gives the higher inductance. Multiplying a rod-wound solenoid by a ferrite's full permeability of 2,000 therefore overstates it badly. This calculator applies the material permeability only to a closed toroid, and for everything else, rods, E-cores, pot cores and gapped cores, it uses the AL value the manufacturer measured on that exact shape.
A toroid of rectangular cross-section follows L = 2N²h ln(d2 ÷ d1) in nanohenries with the height and diameters in centimeters, the air-core form given in the same table. With a core filling the ring, the inductance is that figure times the relative permeability, and the product µr × 2h ln(d2 ÷ d1) is exactly the toroid's equivalent AL.
Permeability of core materials, from air to Metglas
Figures from Wikipedia's permeability table, which warns that the permeability of ferromagnetic materials varies greatly with field strength, composition and fabrication. Ranges are what the table gives, and the maximum values are measured at the stated field.
| Material | Relative permeability | Condition or frequency |
|---|---|---|
| Air | 1.00000037 | treat as 1 |
| Cobalt nickel zinc ferrite | 40-125 | about 2-150 MHz |
| Sendust (Al-Si-Fe powder) | 14-160 | powder core |
| Nickel iron powder | 14-160 | about 50 Hz-2 MHz |
| Molypermalloy powder (MPP) | 14-550 | about 50 Hz-3 MHz |
| Nickel zinc ferrite | 10-2,300 | about 1 kHz-400 MHz |
| Manganese zinc ferrite | 350-20,000 | about 100 Hz-4 MHz |
| Iron, 99.8% pure | 5,000 | maximum |
| Electrical steel | 2,000-38,000 | at 0.002 T and at 1 T |
| Mu-metal | 20,000-50,000 | 20,000 at 0.002 T |
| Nanocrystalline (NANOPERM) | 80,000 | at 0.5 T, 10 kHz |
| Permalloy | 100,000 | at 0.002 T |
| Metglas 2714A, annealed | 1,000,000 | at 0.5 T, 100 kHz |
The ranges are wide on purpose. Two ferrites both called "MnZn" can differ by a factor of fifty, which is why the permeability field has no preset list: the number belongs on the datasheet of the core in your hand.
Eleven coils run through the calculator
Each row is a real input set typed into the form; the result column is what the calculator returns.
| Coil | Input | Result | Note |
|---|---|---|---|
| Air coil | 100 turns, 20 × 50 mm | 67.57 µH | ideal 78.96 µH, ratio 0.856 |
| Short air coil | 10 turns, 20 × 5 mm | 2.817 µH | ideal 7.896 µH, ratio 0.357 |
| Long air coil | 200 turns, 10 × 200 mm | 19.55 µH | ideal 19.74 µH, ratio 0.99 |
| Air coil in inches | 50 turns, 1 × 2 in | 25.81 µH | ideal 31.34 µH |
| Ferrite toroid | 10 turns, 15/25/10 mm, µr 2,000 | 204.3 µH | equivalent AL 2,043 nH |
| Same ring in air | 10 turns, 15/25/10 mm, µr 1 | 102.2 nH | AL 1.022 nH |
| Powder toroid | 30 turns, 15/25/10 mm, µr 75 | 68.96 µH | AL 76.62 nH |
| AL core | 20 turns, AL 250 nH | 100 µH | 40 turns: 400 µH |
| Turns for 100 µH, air | 20 × 50 mm form | 122 turns | exactly 121.66; 122 give 100.6 µH |
| Turns for 1 mH, toroid | 15/25/10 mm, µr 2,000 | 23 turns | 23 give 1.081 mH, 22 give 989 µH |
| Turns for 47 µH, AL core | AL 100 nH | 22 turns | 22 give 48.4 µH, 21 give 44.1 µH |
Patterns hidden in those rows
Reading the numbers you get back
| What you see | What it means |
|---|---|
| Wheeler ÷ ideal above 0.9 | long coil, the textbook formula is nearly right |
| Wheeler ÷ ideal between 0.5 and 0.9 | ordinary proportions, the ideal formula overstates by 11-100% |
| Warning about a fifth of the diameter | outside the formula's range, treat the figure as rough |
| Equivalent AL of a toroid | compare it with the datasheet AL; a large gap means the permeability you typed is not the one the core has |
| Suggested turns and the two values beside it | pick the side of the target your circuit tolerates; the exact count is rarely a whole number |
All results are small-signal values. Core permeability falls toward saturation at high current, and the measured inductance of a finished coil also depends on how tightly and evenly it is wound.
Coil questions answered from the formulas
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