Inductance calculator

Use the calculator

Pick the mode at the top — Solenoid, Toroid, Wire, Brooks, X_L, LC resonance, or Energy. Each mode has its own input set and emits a result with the calculation breakdown plus any accuracy warnings.

The seven inductance formulas

This is the defining equation. A 1-henry inductor carries 1 V across its terminals when current changes at 1 A/s. Doubling the rate of change doubles the voltage. For DC steady current (di/dt = 0), the inductor looks like a wire (zero impedance). For pure AC, the formula transforms into X_L = 2π·f·L.

The classic empirical fit by Harold Wheeler. For coils filled with high-μᵣ material, multiply L by μᵣ — but only when the fill is uniform. For partial cores, the toroid or measured-permeability approach is more honest.

Toroidal cores confine the magnetic flux entirely inside the ring, so they radiate very little EMI and have low leakage inductance — preferred for power supplies, common-mode chokes, and audio transformers. The formula assumes the core cross-section is small relative to the mean radius.

Even a short piece of straight wire has measurable inductance. A 10 cm length of 1 mm-radius wire is about 100 nH at HF — at 1 GHz this is X_L ≈ 0.6 Ω. This is why RF circuits use wide PCB traces, ground planes, and short component leads, and why high-current power-supply layouts pay attention to bus-bar inductance.

The Brooks coil — discovered by H.B. Brooks at the US National Bureau of Standards in 1931 — is the geometry that maximises Q for a given total length of wire. Inner radius = a/2, outer = 3a/2, axial winding length = a. The constant 16.94 nH/(cm·turn²) is derived from Maxwell's mutual-inductance integrals at this exact c = b = a aspect ratio. Used in NBS standard inductors and high-Q laboratory coils.

At f₀, X_L = X_C and the reactive elements cancel — the only impedance left is the series resistance R. Quality factor Q = ω₀·L / R determines bandwidth: a Q of 100 means the resonance peak is f₀ ± f₀/200 wide.

Worked examples

Example 1 — RF coil for the AM broadcast band

Design an air-core inductor that resonates with a 365 pF tuning capacitor at 1 MHz (centre of the AM band). From f₀ = 1/(2π·√(LC)) we need L ≈ 70 µH. Picking r = 10 mm and ℓ = 50 mm in Wheeler's formula gives:

Example 2 — Power-supply choke for a 1 A buck converter

A 100 kHz buck converter needs about 100 µH at 1 A peak. A toroidal iron-powder core (Carbonyl E, μᵣ ≈ 35) with cross-section 25 mm² and mean radius 12 mm needs how many turns?

Solving the toroid formula: L = μ₀ · 35 · N² · 25e⁻⁶ / (2π · 0.012) → N² = L · 2π · 0.012 / (μ₀ · 35 · 25e⁻⁶) = 100e⁻⁶ · 0.0754 / (1.10e⁻⁹) ≈ 6850, so N ≈ 83 turns. Stored energy at peak current: E = ½ · 100e⁻⁶ · 1² = 50 µJ — small enough that no flyback diode is needed for the freewheel path; the synchronous-rectifier MOSFET handles it directly.

How to compute inductance, step by step

1. Pick the geometry that matches your physical layout. Single-layer cylinder of wire on a former → solenoid (Wheeler). Closed ring of ferrite or iron-powder → toroid. A length of wire alone (PCB trace, bus bar, ground strap) → straight wire. Compact multilayer air-coil designed for max-Q resonance work → Brooks coil. The geometry decides the formula.
2. Choose the core material. Air or plastic former (μᵣ = 1) for RF and high-Q work. Ferrite (μᵣ 100–6 000) for SMPS, EMI filters, and power chokes. Iron powder (μᵣ ~35) for high-current DC chokes that resist saturation. Silicon-steel laminated (μᵣ ~1 500) for line-frequency 50 / 60 Hz transformers. Permalloy (μᵣ ~50 000) for audio and instrumentation.
3. Enter the dimensions in any consistent unit. For the solenoid: radius from coil axis to the centre of the wire, length is the winding length, N is total turn count. For the toroid: cross-section area of the core, mean radius (centre of toroid to centre of core). For straight wire: length and wire radius (not diameter). For Brooks: just the mean radius and turn count — geometry is fixed at c = b = a.
4. Read the inductance and any warnings. The calculator returns L in the most natural unit (H / mH / µH / nH). It also flags violations of the formula's assumptions: solenoid is short (ℓ < 0.8 r), wire is too thick (ℓ/r < 100), toroid cross-section is too large, core may saturate. Treat warnings as accuracy bounds, not blockers — the result is still useful for first-pass design.
5. Switch to X_L mode and check at your operating frequency. Inductive reactance X_L = 2π·f·L tells you the impedance of your inductor at the AC frequency that matters. For a 60 Hz line choke this is hundreds of ohms; for a 1 MHz RF coil, hundreds of kilohms. If X_L is too low, more turns or higher μᵣ; if too high, fewer turns or lower μᵣ. Tune from there.
6. For tuned circuits, switch to LC resonance and verify f₀. Pair the inductor with the capacitance (PCB stray + chosen tuning cap) and confirm f₀ = 1 / (2π·√(LC)) lands on your target. Add the series resistance to read the loaded Q and the bandwidth f₀ / Q. Real-world Q peaks at 100–500 for air coils, 30–100 for ferrite cores at HF, and 10–50 for powdered-iron cores.

Variants and special cases

Multilayer coil

For coils where multiple winding layers stack radially, Wheeler's single-layer formula underestimates L. The multilayer formula (Wheeler 1928 second form) is L (µH) = 0.31·(rN)² / (6r + 9ℓ + 10b), where b is the radial build of the winding stack. Multilayer coils have lower self-resonant frequency because of inter-layer capacitance — typically used at low frequency only.

Air-gap inductor

To prevent core saturation in DC chokes, the magnetic path is interrupted with a small non-magnetic gap (a thin Mylar sheet, ground gap, or distributed gap in iron powder). The gap dominates total reluctance: effective μᵣ ≈ 1/((1/μᵣ_core) + (g/ℓ_path)), where g is gap length and ℓ_path is the magnetic path length. A 0.5 mm gap in a ferrite core with 70 mm path length and μᵣ = 3000 reduces effective μᵣ to about 140 — still useful, but linear in current up to much higher levels.

RF straight-wire inductance

Even a straight piece of wire has self-inductance: L (nH) ≈ 2·ℓ·(ln(2ℓ/r) − 0.75), with ℓ and r in cm. A 10 cm bus bar of 1 mm wire has L ≈ 100 nH — at 1 GHz this is X_L ≈ 0.6 Ω, the reason RF circuits use very wide PCB traces and ground planes for return current.

Mutual inductance and coupling

Two coupled inductors (transformer windings) have a mutual inductance M = k·√(L₁·L₂), where k is the coupling coefficient (0–1). Tightly-coupled transformer pairs reach k ≥ 0.99; loosely-coupled wireless-charging pads sit at k ≈ 0.3–0.5. The mutual term changes the apparent inductance seen by the source — use a network analyser for accurate measurements.

Where inductance matters

Filters

Series inductor + parallel capacitor = low-pass filter. The cut-off frequency is f_c = 1/(2π·√(LC)) — same formula as resonance. Power-supply input filters use this to keep switching noise from radiating back onto the AC line.

Energy storage in switching converters

Buck, boost, and flyback converters store energy in an inductor during the "on" half-cycle and release it during "off". The inductance value sets the ripple current: smaller L = more ripple but smaller physical part. Typical SMPS inductors are 1–100 µH at 100 kHz–1 MHz.

RF tuned circuits

An LC tank in a radio receiver picks one frequency from the antenna. The Q of the tank determines selectivity — how cleanly the receiver rejects adjacent stations. Q values of 100–500 are typical for medium-wave receivers.

Motor windings

Every electric motor is fundamentally a set of inductors. Winding inductance limits how fast current can build up when the rotor turns; this drives the back-EMF and ultimately limits motor speed for a given supply voltage.

Lightning surge protection

A high-current inductor in the supply line slows the rate of rise of a lightning surge enough that the downstream metal-oxide varistor (MOV) has time to clamp. Mains surge protectors combine an MOV with a small series inductor for this reason.

Related concepts on this site

Wheeler's original formula

For a single-layer solenoid, the inductance is given by L = (a·N)² / (9a + 10b), where L is in microhenries, a is the radius and b is the length of the coil, both in inches. The error of this formula does not exceed one per cent when b/a is greater than 0.8.

Frequently asked questions

What is inductance?

Inductance is the property of a conductor that resists changes in current by storing energy in a magnetic field. The unit is the henry (H): 1 H = 1 V·s/A, meaning a current changing at 1 A/s induces 1 V across the inductor. Practical inductors range from picohenries (PCB trace inductance) to henries (line-frequency power inductors and audio chokes).

What is the formula for inductance of a solenoid?

Wheeler 1928 (imperial form) is accurate to ±1 % for ℓ ≥ 0.8 r: L (μH) = (r·N)² / (9r + 10ℓ), with r and ℓ in inches. The metric form uses the same ratio after converting r and ℓ to inches. For a uniformly-filled core, multiply by relative permeability μᵣ. For longer coils (ℓ ≫ r) the simpler formula L = μ₀·μᵣ·N²·A/ℓ applies, where A is the cross-section area.

What is inductive reactance?

Inductive reactance X_L is the AC opposition an inductor presents at frequency f: X_L = 2π·f·L. Unit is ohms. At DC (f = 0), X_L = 0 — the inductor passes DC freely. At high frequency, X_L grows linearly. This is the basis of low-pass filters and choke design: an inductor in series blocks AC, passes DC.

How do I calculate the resonance frequency of an LC circuit?

f₀ = 1 / (2π·√(LC)). At this frequency, X_L = X_C and the LC tank impedance is minimum (series LC) or maximum (parallel LC). A 100 µH inductor with a 100 pF capacitor resonates at about 1.59 MHz — typical AM broadcast band. The Q factor Q = ω₀·L / R determines bandwidth: bandwidth = f₀ / Q.

How much energy does an inductor store?

E = ½·L·I². Energy is stored in the magnetic field; the same energy returns to the source on de-energisation. A 1 H inductor at 10 A stores 50 J — enough to weld a switch shut if interrupted abruptly. Switching circuits use a flyback diode or RC snubber to absorb this energy harmlessly.

Why does a ferrite-core inductor have so much more inductance?

Ferrite has a high relative permeability (μᵣ from 100 to 10 000 typical), so the same number of turns gives 100–10 000 × the air-core inductance. The trade-off is core saturation: above a critical current, the core stops contributing extra flux and the inductance collapses to its air-core value. Power inductors avoid this with iron-powder or distributed-air-gap cores; high-frequency RF chokes use lower-μ ferrite that saturates only at higher currents.

Series vs parallel inductors — which formula?

For inductors with no mutual coupling (separated or perpendicular orientation): series adds like resistors (L_total = L₁ + L₂ + …); parallel adds reciprocally (1/L_total = 1/L₁ + 1/L₂ + …). When inductors are mutually coupled (windings on a common core), include the coupling coefficient k and mutual inductance M = k·√(L₁·L₂) — total inductance changes depending on whether the windings aid or oppose.

Sources and methodology

1. Wheeler, H.A. Simple Inductance Formulas for Radio Coils. Proceedings of the IRE, Vol. 16, No. 10 (October 1928), pp. 1398–1400.
2. Brooks, H.B. Design of Standards of Inductance, and the Proposed Use of Model Reactors. Bureau of Standards Journal of Research, Vol. 7, No. 2 (1931), pp. 289–328.
3. Rosa, E.B. The Self and Mutual Inductances of Linear Conductors. Bulletin of the Bureau of Standards, Vol. 4, No. 2 (1908), pp. 301–344.
4. Terman, F.E. Radio Engineers' Handbook. McGraw-Hill, 1943. Chapter 2 — Coils and inductances.
5. BIPM. The International System of Units (SI), 9th Edition, 2019. μ₀ = 1.25663706212 × 10⁻⁶ H/m.
6. IEEE. IEEE Std 100 — The Authoritative Dictionary of IEEE Standards Terms, 7th Edition, 2000. Definitions of inductance, reactance, Q-factor.
7. Snelling, E.C. Soft Ferrites — Properties and Applications, 2nd Ed., Butterworths, 1988. Reference for μᵣ vs frequency in ferrite materials.
8. Magnetics Inc. Powder Core Catalog 2020. Iron-powder and Kool-Mu material datasheets.
9. Steinmetz, C.P. Theory and Calculation of Alternating Current Phenomena, McGraw-Hill, 1897. Foundational treatment of inductive reactance.