Calculation
L = μ0 · μr · N² · A/l
Inductance grows with the square of turns and linearly with core area and relative permeability.
L = μ0·μr·N²·A/l
Toroidal (ring-core) inductor inductance
Inductance grows with the square of turns and linearly with core area and relative permeability.
MINTSI
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Result
Select a target and calculate.
Understand the inputs
- Inductance L — Inductance of the toroidal coil.
- Turns N — Number of winding turns around the core.
- Core area A — Effective cross-sectional area of the toroidal core.
- Mean magnetic path length l — Mean circumference of the magnetic circuit in the core.
- Relative permeability μr — Core material property; air = 1, ferrite cores for power converters typically a few hundred to several thousand.
Example
100 turns on a ferrite core with a 100 mm² cross-section, 100 mm mean path length and μr = 2,000 give L ≈ 25.1 mH.
Assumptions and limits
Homogeneous core material without an air gap, no saturation and no frequency-dependent core losses.
Understand Toroidal (ring-core) inductor inductance
This calculator finds the inductance of a toroidal coil wound on a ferromagnetic core from turns, core cross-section, mean magnetic path length and relative permeability — or, in reverse, the turns needed for a target inductance.
What does this quantity describe?
A coil with N turns carrying current I stores a linked magnetic flux N·Φ = L·I. For a toroidal coil on a ferromagnetic core with cross-sectional area A and mean magnetic path length l, inductance follows from L = μ0·μr·N²·A/l, where μ0 is the magnetic constant and μr is the core material's relative permeability.
The toroidal core acts like a closed channel for magnetic flux: a material with high permeability μr is like a channel with very low flow resistance, letting the same flux pass with much less magnetic driving force — which is why a few turns on a ferrite core give the same inductance as far more turns on air (μr = 1).
Formula and variables
L = μ0 · μr · N² · A/l
L = μ0·μr·N²·A/lN = √(L·l/(μ0·μr·A))
| Symbol / input | Meaning |
|---|---|
| Inductance L | Inductance of the toroidal coil. |
| Turns N | Number of winding turns around the core. |
| Core area A | Effective cross-sectional area of the toroidal core. |
| Mean magnetic path length l | Mean circumference of the magnetic circuit in the core. |
| Relative permeability μr | Core material property; air = 1, ferrite cores for power converters typically a few hundred to several thousand. |
Choose the inputs correctly
Turns N, core area A, mean magnetic path length l and the core material's relative permeability μr set the inductance. A and l are usually given on the core datasheet; μr is strongly material- and frequency-dependent and should be taken from the datasheet for the intended frequency range.
How to use the calculator
Select the target quantity. For inductance, enter N, A, l and μr. For a sizing question — how many turns a target inductance needs — choose N as the target and enter the desired inductance L.
Worked example
100 turns on a ferrite core with a 100 mm² cross-section, 100 mm mean path length and μr = 2,000 give L = 4π·10⁻⁷ · 2,000 · 100² · 10⁻⁴ m² / 0.1 m ≈ 25.1 mH.
Understand the result and units
An inductance of 25.1 mH from just 100 turns shows the strong effect of the core material: the same coil without a core (μr = 1) would give only about 12.6 µH — the ferrite core raises inductance here by the factor μr = 2,000.
Core area is usually given in mm², magnetic path length in mm, and the resulting inductance falls in the microhenry-to-henry range depending on core size.
What is a core's AL value?
Core manufacturers often additionally state an AL value for toroidal cores, usually in nH per turn squared — it bundles μ0·μr·A/l into a single figure, so inductance can be computed directly from L = AL·N² without entering core area, path length and permeability separately. The AL value is only valid over the drive level and frequency range the manufacturer specifies, since it already incorporates the material-dependent permeability.
Typical applications
The relationship is used to design chokes, EMI filters (common-mode chokes), transformers and storage inductors in switch-mode power supplies, and to pre-estimate the turns needed for a given core and target inductance.
Assumptions, limits and common mistakes
The model assumes homogeneous core material without an air gap and a uniform field distribution along the mean path length. Core saturation at high currents, the pronounced nonlinearity of a real ferrite's B-H curve, and frequency-dependent eddy-current and hysteresis losses are not included; at high frequencies or high currents, real inductance therefore deviates noticeably from this linearized model.
Common mistake: A common mistake is confusing the effective magnetic path length l with the ring's outer geometric circumference — core manufacturers state l as a datasheet value that can differ from the actual geometry and field distribution in the core. It is also easy to treat μr as a fixed constant across the whole operating range, even though it varies noticeably with frequency, temperature and drive level in ferrites.
Frequently asked questions
Why does inductance grow with the square of the turn count?
Because each added turn both increases the magnetic flux produced and increases the number of turns linked to that flux — the two effects multiply, so inductance scales with N².
What's the difference between an air-core coil and a ferrite-core coil?
An air-core coil (μr = 1) has low inductance per turn but very low core losses and is immune to saturation; a ferrite core raises inductance by the factor μr but introduces saturation and frequency-dependent core losses.
What is the AL value?
It is a figure given by the core manufacturer that bundles μ0·μr·A/l, so inductance can be computed directly from L = AL·N².
What effects does this model not cover?
Core saturation at high currents, the nonlinear B-H curve of a real ferrite, and frequency-dependent eddy-current and hysteresis losses are not included.
How many turns do I need for a target inductance?
Choose turns N as the target quantity, enter the desired inductance L along with the core area, path length and permeability of the intended core; the calculator solves the formula for N.