Inductance Calculator: Compute L for Toroids, Solenoids, PCB TracesInductance is a fundamental property of electrical conductors and components that quantifies their ability to store energy in a magnetic field when an electric current flows. Engineers, hobbyists, and students often need to estimate inductance quickly for circuit design, electromagnetic compatibility (EMC) analysis, or component selection. This article explains the basics of inductance, presents common formulas for solenoids, toroids, and PCB traces, discusses practical factors that affect accuracy, and shows how an inductance calculator can streamline the process.
What is inductance?
Inductance (L) measures the voltage induced in a conductor due to the time-varying current through itself (self-inductance) or another nearby conductor (mutual inductance). It is defined by the relationship:
V = L (dI/dt)
where V is the induced voltage, I is current, and dI/dt is the rate of change of current. Inductance is measured in henries (H). Practical inductances range from picohenries (pH) for small PCB traces to millihenries (mH) or henries for larger coils and inductors.
Why use an inductance calculator?
- Saves time versus manual calculation.
- Reduces errors by using correct geometric factors and unit conversions.
- Compares multiple geometries quickly (solenoid vs toroid vs trace).
- Helps iterate design parameters (turns, spacing, core material).
- Useful during PCB layout, filter design, and antenna matching.
Key parameters that determine inductance
- Geometry: length, diameter, number of turns, spacing, cross-sectional area.
- Core material: air, ferrite, powdered iron — relative permeability (μr) strongly affects L.
- Winding distribution: single-layer, multi-layer, close-wound, spaced turns.
- Frequency effects: skin effect and proximity effect increase AC resistance and alter effective inductance at high frequencies.
- Mutual coupling and nearby conductors: nearby metal or traces change the magnetic path and L.
Solenoids
A solenoid is a coil of wire wound in a helix. For a long solenoid (length >> diameter) with N turns, cross-sectional area A, and length l, an approximate inductance is:
L ≈ μ0 μr N^2 A / l
where μ0 = 4π × 10^-7 H/m (permeability of free space) and μr is the relative permeability of the core (μr = 1 for air).
More accurate formulas include end correction factors and account for non-ideal aspect ratios. For single-layer short coils, Wheeler’s approximate formula is widely used (practical and simple):
For a single-layer air-core solenoid (units: inches): L(μH) ≈ (r^2 N^2) / (9r + 10l)
where r = coil radius (in), l = coil length (in), N = number of turns. For metric units (mm), an alternative Wheeler form is: L(μH) ≈ (0.0002 N^2 r^2) / (9r + 10l) (r and l in mm)
Use these formulas for quick, reasonably accurate estimates (typically within 5–10% for well-behaved coils).
Toroids
Toroidal coils confine the magnetic flux within a doughnut-shaped core, which reduces external magnetic fields and improves coupling to the core. For a toroid with mean radius R, cross-sectional area A, N turns, and core relative permeability μr:
L ≈ (μ0 μr N^2 A) / (2π R)
This assumes the magnetic path is mostly inside the core (good for high μr cores) and that the core cross-section is small compared to the mean radius. More precise calculations account for non-uniform flux distribution and gapped cores.
Wheeler also provides a simple empirical formula for toroidal inductors with circular cross-sections (metric-friendly variant available in many references). For practical work, the core manufacturer’s datasheet often provides inductance per turn or AL value (nH/turn^2), which is the easiest way to compute L:
L = AL × N^2
where AL is given in nH/turn^2; convert units as needed.
PCB traces (microstrip, stripline, and loop inductance)
PCB inductance is often the dominant factor at high frequencies for short connections and traces. Two common inductance types to consider are:
- Trace inductance per unit length (useful for long traces and transmission-line behavior).
- Loop inductance for signal-return loops (important for power distribution and EMC).
Approximate formulas:
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Inductance per unit length of a straight round conductor in free space: L’ ≈ (μ0 / 2π) [ln(2l/r) – 1] This is not often used directly for PCB traces because traces are flat and near dielectric and reference planes.
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Empirical estimate for a PCB trace over a ground plane (microstrip-like behavior) — for quick approximation, use layout tools or field solvers. For a narrow trace of width w and height above ground h, inductance per unit length (approx): L’ ≈ μ0 h / w (order-of-magnitude estimate) This is crude; better accuracy requires electromagnetic simulation.
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Loop inductance estimate (useful for short loop areas): L ≈ μ0 * (perimeter) * [ln(2 * perimeter / conductor_width) – 1] / π For very small loops, use approximations or measurement; layouts with minimized loop area reduce inductance and EMI.
Practical tip: placing a solid reference plane (ground) close to signal traces drastically lowers loop inductance and EMI.
Core materials and AL values
Manufacturers publish AL values for ferrite and powdered-iron cores. AL relates inductance to turns:
L (nH) = AL (nH/turn^2) × N^2
AL already includes geometry and permeability, so it’s the simplest method for toroid designs: pick a core with known AL, choose N, compute L, and check for core saturation using modulation of current and core cross-sectional area.
Frequency effects and losses
At higher frequencies:
- Skin effect concentrates current near conductor surfaces, reducing effective cross-section and increasing resistance.
- Proximity effect from neighboring conductors changes current distribution and can reduce inductance slightly.
- Dielectric and core losses (loss tangent, hysteresis) cause energy dissipation and an effective series resistance (ESR).
- Parasitic capacitance between turns forms a self-resonant frequency (SRF); above SRF the coil behaves capacitively.
An inductance calculator should warn about SRF and provide AC models (L with series R and parallel C) for RF work.
How an inductance calculator works — features to expect
- Geometry inputs: N, coil length, wire diameter, coil inner/outer diameters, spacing.
- Core selection: air, ferrite, powdered iron, with μr or AL value input.
- Unit conversion and sensible defaults.
- Multiple formula options (Wheeler, long-solenoid, toroid formula, AL-based).
- Frequency-dependent options: compute skin depth, SRF estimate, and effective AC resistance.
- Visualization: coil dimensions, winding cross-section, and magnetic path.
- Export results and comparison mode for different geometries.
Worked examples
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Solenoid (air): N = 50 turns, radius r = 10 mm, length l = 40 mm, μr = 1. Use Wheeler approximate formula to estimate L; if you need a numeric result, an inductance calculator will convert units and apply the formula.
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Toroid (ferrite): AL = 500 nH/turn^2, N = 10 turns. L = 500 × 10^2 = 50,000 nH = 50 μH.
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PCB loop: small rectangular loop 20 mm × 10 mm made from 1 mm trace at 1 mm above ground. A field solver or calculator will estimate loop inductance on the order of a few nH; minimizing loop area reduces L proportionally.
Accuracy and validation
- For conceptual design and early-stage estimates: formulas above are usually sufficient.
- For final designs, especially at RF or where tight tolerances matter: use a field solver (FEM), measure prototypes with an LCR meter, and consult core datasheets.
- Compare multiple formulas and consider the AL method when using commercial cores.
Summary
An inductance calculator that supports solenoids, toroids, and PCB traces speeds design and reduces guesswork by applying geometry-specific formulas and core data (AL values). For the best results, combine calculator estimates with manufacturer data, EM simulation, and physical measurement where accuracy matters.
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