Coil Inductance Formulas

Air-core solenoid (Wheeler)

L (µH)

= (r²·N²) / (9r + 10ℓ) — all in inches

SI form

L = μ₀ · N² · A / ℓ (for long solenoid)

r

Coil radius

ℓ

Coil length

N

Number of turns

Multilayer (Wheeler)

L (µH)

= (0.8·a²·N²) / (6a + 9ℓ + 10c)

a

Mean radius (in)

ℓ

Winding length (in)

c

Winding depth (in)

Toroid

L (H)

= μ₀ μ_r · N² · A_e / ℓ_e

A_e

Effective cross-section area

ℓ_e

Mean magnetic path length

A_L

Core-specific factor: L (nH) = A_L · N²

Design notes

• Adding ferromagnetic core increases L by μ_r (hundreds to thousands for ferrite).
• Saturation current I_sat: above it, the core loses permeability and L drops sharply.
• For RF, air-core is preferred for highest Q and zero saturation.
• Self-resonant frequency (SRF) is where parasitic capacitance makes the coil look like a capacitor.