How accurate is this formula?

Very accurate for long solenoids where length is at least 10 times the diameter. For short coils the formula overestimates because it assumes uniform field. Correction factors exist for short coils.

What permeability should I use for air?

Use 1 for air vacuum or any non-magnetic material like plastic wood or aluminium. Ferrite cores range from 100 to 10000. Iron cores range from 1000 to 10000.

How do I increase inductance without adding turns?

Use a higher permeability core material. A ferrite core with mu r of 2000 multiplies the inductance by 2000 compared to air. Alternatively increase the diameter or decrease the length.

Solenoid Inductance Calculator – Turns, Core, Dimensions

L = μ₀μ_rN²A / l

Single-layer solenoid. Find inductance from geometry, or turns from target L.

N Number of Turns

d Coil Diameter

l Solenoid Length

μ_r Relative Permeability (1 = air)

L_t Target Inductance (for Find Turns mode)

Analysis Results

Inductance
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Turns
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Cross-Section

L per Turn

μ_r

Length/Diameter

Solenoid Inductor Geometry

A solenoid is a coil of wire wound in a helix. Its inductance depends on the number of turns squared, the cross-sectional area, the length, and the core material. The formula assumes a long solenoid (length >> diameter).

d — Coil diameter. Area = π(d/2)². Larger diameter = more inductance.
l — Solenoid length. Longer solenoid = less inductance (field is spread thinner).
N — Number of turns. Inductance scales with N² — doubling turns quadruples L.
μ_r — Relative permeability of core material. Air = 1. Ferrite = 100–10000. Iron = 1000–10000.

Solenoid Inductance Calculator

A solenoid is a coil of wire wound in a helix. Its inductance depends on geometry (turns, diameter, length) and the core material. This calculator finds the inductance from physical dimensions, or reverses to find how many turns you need for a target inductance.

The Inductance Formula

L = μ&sub0;μ_rN²A / l

μ&sub0; = 4π × 10&supmin;&sup7; H/m (permeability of free space)
μ_r = relative permeability of core (1 for air)
N = number of turns
A = cross-sectional area = π(d/2)²
l = solenoid length

Inductance scales with N² — doubling the turns quadruples the inductance. It also scales linearly with core permeability and area, and inversely with length. For the magnetic field this solenoid produces, see the Solenoid Magnetic Field Calculator.

Core Materials and Permeability

Air / vacuum: μ_r = 1. Lowest inductance. No saturation. Used for RF coils, high-frequency filters.

Ferrite: μ_r = 100–10000. Moderate inductance boost. Low losses at high frequency. Used for SMPS inductors, EMI filters, transformers.

Iron powder: μ_r = 10–100. Distributed gap. Soft saturation. Used for power inductors, DC chokes.

Laminated iron: μ_r = 1000–10000. Highest inductance. Used for mains-frequency transformers, relays, solenoids.

Air-Core (100 turns / 10mm / 50mm)

A = π(5 mm)² = 78.5 mm²
L = 4π×10&supmin;&sup7; × 1 × 100² × 78.5×10&supmin;&sup6; / 0.05
L = 19.7 µH

19.7 µH for a 100-turn air-core coil. Typical for RF applications. For the energy this inductor stores, see the Inductor Energy Calculator.

Ferrite Core (μ_r = 2000)

Same geometry, 50 turns instead of 100:
L = 4π×10&supmin;&sup7; × 2000 × 50² × 78.5×10&supmin;&sup6; / 0.03
L = 16.4 mH

The ferrite core boosts inductance by 2000×, and with only 50 turns we get 16.4 mH — 830× more than the 100-turn air core. This is why ferrite cores are used everywhere in power electronics.

Iron Core (μ_r = 5000)

200 turns, 25mm diameter, 100mm length:
L = 4π×10&supmin;&sup7; × 5000 × 200² × 491×10&supmin;&sup6; / 0.1
L = 1.23 H

Over 1 henry. This is relay and transformer territory. For a closed magnetic path (no air gap, even higher L), see the Toroid Inductor Calculator.

Reverse: Find Turns for 100 µH

N = √(L × l / (μ&sub0;μ_r × A))
For air core, 10mm diameter, 50mm length:
N = √(100×10&supmin;&sup6; × 0.05 / (4π×10&supmin;&sup7; × 78.5×10&supmin;&sup6;))
N = 226 turns

Design Tips

Length/diameter ratio: The formula is most accurate when l/d > 10. For short coils, use Nagaoka's correction factor.

Multi-layer coils: This formula assumes a single-layer winding. Multi-layer coils have higher inductance per turn but also higher parasitic capacitance.

Core saturation: The formula assumes constant μ_r. At high currents, the core saturates and μ_r drops dramatically, reducing inductance. Check the core's B-H curve.

Frequently Asked Questions

Why does inductance scale with turns squared?

Each turn produces flux and links flux from every other turn. Doubling turns doubles the flux produced and doubles the flux linked to each turn — giving 2 × 2 = 4 times the inductance.

What is a realistic permeability for ferrite?

Ranges from 100 (high-frequency NiZn) to 10000 (high-permeability MnZn). Common power ferrites are 2000–3000. Always check the datasheet — permeability varies with frequency, temperature, and DC bias.

Does this formula work for toroidal cores?

Approximately, using the mean magnetic path length as l and the core cross-section as A. For precise toroid calculations with the specific AL value, use the Toroid Inductor Calculator.

How do I account for a gapped core?

A gap reduces the effective permeability. The effective μ_r is approximately l_core / (l_core/μ_r + l_gap). Gaps are deliberately introduced to prevent saturation and stabilise inductance vs current.

Last updated: March 2026