Coils store energy in a magnetic field
Wind wire into a coil and pass a current through it: a magnetic field builds up inside, and the coil resists any change in that current. This property is inductance, measured in henries (H). Coils appear in radio tuners, filters, switching power supplies, relays and wireless chargers. For a simple single-layer coil, two formulas give a good estimate.
The long-solenoid formula
L = μ₀ × μr × N² × A ÷ l
- μ₀ = 4π × 10⁻⁷ H/m is the permeability of free space.
- μr is the relative permeability of the core: 1 for air or plastic, much more for ferrite or iron.
- N is the number of turns.
- A is the cross-sectional area, π × d² ÷ 4.
- l is the length of the winding.
The formula assumes the field is uniform inside and zero outside, which is true only when the coil is much longer than its diameter. For shorter coils it overestimates the inductance.
Wheeler’s formula
In 1928 the American engineer Harold Wheeler published an empirical formula for single-layer air-core coils that works for short coils too:
L (µH) = d² × N² ÷ (18d + 40l), with d and l in inches.
It is accurate to about 1% when the length is more than 0.4 times the diameter.
Worked example
A coil of 200 turns wound on a form 1 inch in diameter and 2 inches long, with an air core and 2 A of current:
- Long-solenoid formula: about 501 µH.
- Wheeler’s formula: 1² × 200² ÷ (18 + 80) = 408 µH. Because the coil is only twice as long as it is wide, Wheeler’s value is the more realistic one.
- Magnetic field inside: μ₀ × 200 × 2 ÷ 0.0508 m = about 9.9 mT.
- Stored energy: ½ × L × I² = about 1.0 mJ.
What changes the inductance
| Change | Effect on inductance |
|---|---|
| Double the turns, same length | × 4 |
| Double the diameter | about × 4 for a long coil |
| Double the length, same turns | about ÷ 2 |
| Add a ferrite core (μr = 100) | up to × 100 |
Cores and saturation
A ferrite or iron core concentrates the field and multiplies the inductance, but only up to a point. Above a certain current, the core saturates and the inductance falls sharply. A rod-shaped core that does not close the magnetic path also gives a much smaller boost than its material permeability suggests. Manufacturers therefore give an A_L value, in nanohenries per turn squared, for each core: L = A_L × N².
Practical tips
- Multi-layer coils, toroids and flat spiral coils need other formulas.
- At high frequencies, the capacitance between turns and the skin effect change the behavior; a coil resonates on its own at its self-resonant frequency.
- Measure the finished coil with an LCR meter if the value matters: winding tightness and wire thickness shift the result by several percent.