Engineering · Electrical Engineering · Circuit Analysis
Inductance Calculator
Calculates the inductance of a single-layer air-core solenoid coil using its physical dimensions and number of turns.
Calculator
Formula
L is the inductance in henries (H); \mu_0 is the permeability of free space (4\pi \times 10^{-7} \text{ H/m}); N is the number of turns (dimensionless); A is the cross-sectional area of the coil in square metres (m^2), calculated as \pi r^2 where r is the coil radius; l is the length of the coil in metres (m). This formula applies to a tightly wound single-layer air-core solenoid where the length is significantly greater than the radius.
Source: Hayt & Kemmerly, Engineering Circuit Analysis, 8th Edition; IEEE Std 393-1991 — Standard for Test Procedures for Magnetic Cores.
How it works
Inductance (L) is the property of an electrical conductor that causes it to oppose changes in current by storing energy in a surrounding magnetic field. When current flows through a coiled conductor, each turn contributes a magnetic flux that links with all other turns, producing a cumulative self-inductance effect. The greater the number of turns or the larger the coil cross-section, the stronger the magnetic field and the higher the inductance. Conversely, a longer coil reduces inductance because the same number of turns is spread over a greater length, weakening the average flux linkage per turn.
The formula used in this calculator is the classical solenoid inductance equation: L = \(\mu_0 N^2 A / l\), where \(\mu_0 = 4\pi \times 10^{-7}\) H/m is the permeability of free space, N is the total number of turns, A is the circular cross-sectional area of the coil (\(\pi r^2\)), and l is the axial length of the winding. This equation is derived directly from Ampere's Law and the definition of magnetic flux linkage, and it assumes a uniform winding density, an air (or vacuum) core, and a coil whose length is much greater than its diameter — the classic thin solenoid approximation. For coils where the length is comparable to or shorter than the diameter, the Wheeler short-coil formula provides improved accuracy.
Practical applications span a vast range of disciplines. In RF circuit design, air-core inductors with carefully controlled dimensions are used to form LC tank circuits for frequency selection in radios, antennas, and oscillators. In power electronics, inductors are key components in buck and boost converters, where the inductance value determines ripple current and switching performance. Motor drive engineers rely on inductor calculations when designing input filters to suppress electromagnetic interference (EMI). Even in sensor technology, coil inductance forms the basis of inductive proximity sensors and metal detectors, where changes in the surrounding permeability shift the inductance and trigger detection logic.
Worked example
Suppose you are designing an air-core coil for a 1 MHz LC oscillator and need to know its inductance. The coil has the following specifications:
- Number of turns (N): 100
- Coil radius (r): 10 mm = 0.01 m
- Coil length (l): 50 mm = 0.05 m
Step 1: Calculate the cross-sectional area.
A = \(\pi r^2\) = \(\pi \times (0.01)^2\) = \(\pi \times 0.0001\) ≈ 3.1416 × 10⁻⁴ m²
Step 2: Apply the solenoid inductance formula.
L = \(\mu_0 N^2 A / l\)
L = \((4\pi \times 10^{-7}) \times (100)^2 \times (3.1416 \times 10^{-4}) / 0.05\)
L = \((1.2566 \times 10^{-6}) \times 10{,}000 \times (3.1416 \times 10^{-4}) / 0.05\)
L = \((1.2566 \times 10^{-6} \times 3.1416 \times 10^{-3}) / 0.05\)
L = \(3.9478 \times 10^{-9} / 0.05\)
L ≈ 78.96 nH ≈ 0.0790 μH
Step 3: Interpret the result.
A 0.079 μH (79 nH) inductor paired with an appropriate capacitor can form a resonant LC circuit. For a 1 MHz resonance frequency, the required capacitor would be approximately C = 1/(4π²f²L) ≈ 321 pF. This is a practical value achievable with standard capacitors, confirming the coil geometry is suitable for the application.
Limitations & notes
The formula L = \(\mu_0 N^2 A / l\) assumes an ideal, infinitely long solenoid, which means it is most accurate when the coil length is at least three to five times greater than its diameter. For short or wide coils — where the length-to-diameter ratio approaches 1 or less — this formula can overestimate inductance by 20% or more, and the Wheeler short-coil approximation (L = (r²N²)/(9r + 10l) in inch units) should be used instead. The calculator also assumes a single-layer winding with no core material, so it does not account for ferromagnetic or ferrite cores, which multiply inductance by the relative permeability \(\mu_r\) of the core material. Stray capacitance between turns, winding resistance (which influences the quality factor Q), and skin effect losses at high frequencies are not captured by this model. Finally, mutual inductance between adjacent coils and proximity effects in densely wound windings are outside the scope of this single-component calculation.
Frequently asked questions
What is inductance and why does it matter in circuit design?
Inductance is the ability of a coil or conductor to store energy in a magnetic field and oppose changes in current flow. It is measured in henries (H). In circuit design, inductance is critical for filtering, energy storage in power converters, impedance matching in RF systems, and forming resonant circuits with capacitors. Getting the inductance value right ensures a circuit operates at the correct frequency or efficiently transfers power without excessive ripple.
What is the Wheeler formula and when should I use it instead?
The Wheeler formula for short coils — L = (r²N²)/(9r + 10l) with dimensions in inches and inductance in microhenries — is a well-known empirical approximation that remains accurate to within 1% for coils where the length-to-diameter ratio is between 0.4 and 4. You should use the Wheeler formula when designing coils that are relatively short and fat compared to a long, thin solenoid, which is very common in practice. The standard solenoid formula used in this calculator is better suited to elongated coils with a high length-to-diameter ratio.
How does adding a ferrite core change the inductance?
Inserting a ferrite or iron core inside an air-core solenoid increases its inductance by a factor equal to the relative permeability (\(\mu_r\)) of the core material. For example, a ferrite core with \(\mu_r\) = 100 would increase the inductance by approximately 100 times. To account for a core, multiply the calculated air-core inductance by \(\mu_r\). However, ferrite cores also introduce nonlinearity at high flux densities and frequency-dependent losses, which must be considered in high-power or high-frequency designs.
How does inductance change with the number of turns?
Inductance scales with the square of the number of turns (N²). Doubling the number of turns — while keeping the coil radius and length constant — increases the inductance by a factor of four. This is because both the magnetic field strength and the number of flux linkages increase with N, so the total flux linkage (and hence inductance) grows as N². This quadratic relationship makes the number of turns the most powerful single parameter for tuning inductance in a coil design.
What units are used and how do I convert between them?
Inductance is measured in henries (H). In practical electronics, inductors are commonly specified in millihenries (mH = 10⁻³ H), microhenries (μH = 10⁻⁶ H), or nanohenries (nH = 10⁻⁹ H). This calculator outputs results in both μH and nH for convenience. RF and high-frequency inductors are typically in the nH to low-μH range, while power supply inductors are usually in the μH to mH range, and audio or low-frequency transformers can reach hundreds of millihenries or even several henries.
Last updated: 2025-01-15 · Formula verified against primary sources.