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Inductor Design Calculator

Inductance equals permeability times turns squared times area divided by coil length

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Solenoid Inductance

The solenoid (cylindrical coil) formula calculates inductance from permeability, number of turns, cross-sectional area, and coil length. Inductance scales with the square of the number of turns.

L = μN²A / l

Flat Spiral Air Core Coil (Wheeler’s Formula)

Wheeler’s approximation for a flat (pancake) spiral coil uses mean radius, turns, and coil depth. It is widely used for PCB trace inductors and RFID antenna coils. All dimensions are in meters and the result is in henries.

L = r²n² / ((2r + 2.8d) × 10⁵)

How It Works

This calculator covers two inductor geometries. The solenoid (cylindrical coil) formula uses permeability, turns, cross-sectional area, and coil length. The flat spiral air core coil uses Wheeler’s approximation with mean radius, turns, and coil depth (winding width). Both let you solve for any variable.

Example Problem

Design a 1 mH air-core solenoid with 200 turns and a 0.001 m² cross-section. What coil length is needed?

  1. μ for air ≈ 4π × 10⁻⁷ = 1.2566 × 10⁻⁶ H/m
  2. l = μN²A / L = (1.2566e−6 × 200² × 0.001) / 0.001
  3. l = (1.2566e−6 × 40,000 × 0.001) / 0.001 = 0.0503 m (5 cm)

Flat spiral example: A coil with mean radius r = 0.025 m, 10 turns, and depth d = 0.01 m gives L = 0.025² × 100 / ((0.05 + 0.028) × 10⁵) ≈ 8.013 × 10⁻⁶ H (8 μH).

When to Use Each Variable

  • Solve for Solenoid Inductance — when you know the core permeability, number of turns, cross-sectional area, and coil length, e.g., designing a filter inductor for a power supply.
  • Solve for Permeability — when you have a measured inductance and coil geometry and want to determine the core material's permeability.
  • Solve for Number of Turns — when you need a target inductance from a given core and want to know how many turns to wind.
  • Solve for Spiral Inductance — when designing a flat spiral coil (PCB trace, RFID antenna) and need to estimate inductance from coil dimensions.

Key Concepts

Inductance measures a coil's ability to store energy in a magnetic field. For solenoids, inductance scales with the square of the number of turns because each turn both creates and links flux. Wheeler's approximation handles flat spiral (pancake) coils where the standard solenoid formula is inaccurate. Core material permeability dramatically affects inductance — iron cores can increase it by 1,000x or more over air.

Applications

  • Power electronics: designing filter inductors for switching regulators and DC-DC converters
  • RF engineering: tuning coils for radio receivers and impedance matching networks
  • PCB design: estimating trace inductance for RFID antennas and wireless charging coils
  • Transformer design: calculating primary and secondary winding inductance for power transformers

Common Mistakes

  • Applying the solenoid formula to short, wide coils — it assumes length >> diameter; use Nagaoka's correction or Wheeler's formula for short coils
  • Forgetting that core permeability is not constant — ferromagnetic cores saturate at high flux densities, reducing effective inductance
  • Mixing unit systems — the solenoid formula requires consistent SI units (henries, meters, H/m) or the result will be orders of magnitude off

Frequently Asked Questions

Why does inductance scale with turns squared?

Each turn of wire both creates and links magnetic flux. Doubling the turns doubles the flux created and doubles the flux linked, giving a quadratic (N²) relationship.

What is the permeability of free space?

μ₀ = 4π × 10⁻⁷ H/m ≈ 1.2566 × 10⁻⁶ H/m. Iron cores have relative permeabilities of 1,000–10,000, dramatically increasing inductance.

When is this solenoid formula not accurate?

It assumes a long, tightly wound single-layer coil. For short coils (length < 2× diameter), toroidal shapes, or multi-layer windings, use Wheeler’s approximation or Nagaoka’s correction factor.

What is Wheeler’s formula for a flat spiral coil?

Wheeler’s approximation for a flat spiral air-core coil is L = r²n² / ((2r + 2.8d) × 10⁵), where r is the mean coil radius, n is the number of turns, and d is the coil depth (winding width). It is widely used for PCB inductors and RFID antennas where the coil is wound in a single plane.

What are common applications of inductors?

Inductors are used in power supply filters, RF circuits, transformers, impedance matching networks, and energy storage. Switching regulators depend on inductors to smooth output current.

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