Solenoid Inductance Calculator
Enter the solenoid geometry and core material to find inductance (L), the magnetic field inside the coil (B), stored energy (U), and inductive reactance (X_L). Use the Solve for selector to reverse-calculate turns, length, radius, or core permeability from a target inductance. All values update as you type.
Formula
Worked example
A 200-turn coil, radius 15 mm, length 100 mm, air core: A = pi x (0.015 m)^2 = 7.069e-4 m2. L = 4pi x 10^-7 x 200^2 x 7.069e-4 / 0.10 = 354 uH. At 2 A: B = 4pi x 10^-7 x (200/0.1) x 2 = 5.03 mT, U = 0.5 x 354e-6 x 4 = 0.708 mJ, X_L at 60 Hz = 2pi x 60 x 354e-6 = 0.133 Ohm.
What is solenoid inductance?
A solenoid is a tightly wound helical coil of wire. When current flows through it, the overlapping loops create a strong, nearly uniform magnetic field along the central axis. Self-inductance (L) is the proportionality constant between the changing current and the voltage it induces in the same coil: V = -L x (dI/dt). A higher inductance means a larger opposition to current change, which is why inductors are used to filter signals, store energy in switching regulators, and limit fault currents. Inductance depends entirely on geometry and core material, not on the current itself, provided the core stays below magnetic saturation.
The solenoid inductance formula explained
For an ideal solenoid (length much greater than radius) the self-inductance is L = mu_0 x mu_r x N^2 x A / l, where mu_0 = 4pi x 10^-7 H/m is the permeability of free space, mu_r is the relative permeability of the core (1 for air), N is the number of turns, A is the bore cross-sectional area (pi x r^2 for a circular bore), and l is the axial winding length. Inductance scales with the square of turns: doubling N gives four times the inductance. It scales linearly with core permeability and bore area, and inversely with length. The "n-squared" dependence makes turns count the most powerful single knob for designers.
Magnetic field, stored energy, and inductive reactance
The magnetic field inside the solenoid is B = mu_0 x mu_r x n x I, where n = N/l is the winding density in turns per metre and I is the current. Stored magnetic energy is U = (1/2) x L x I^2, analogous to kinetic energy (1/2 x m x v^2) with inductance playing the role of inertia. Inductive reactance X_L = 2 x pi x f x L is the AC impedance the coil presents at frequency f - it rises linearly with frequency, so inductors pass low-frequency currents but block high-frequency ones. This is the basis of low-pass LC filters and RF choke design.
Practical design limits and corrections
The formula above is accurate to a few percent when the winding length exceeds roughly three times the coil diameter. Shorter, fatter coils need the Nagaoka correction factor, which reduces the effective inductance by up to 30% for a single-layer coil whose length equals its diameter. Iron and ferrite cores saturate at a maximum flux density (around 0.3-0.4 T for Mn-Zn ferrite, 1.5-2 T for silicon steel): above saturation, effective permeability drops sharply and inductance collapses. High-frequency windings suffer from skin effect (current pushed to wire surface) and proximity effect (adjacent turns deflecting each other's current), both of which raise winding resistance and reduce the quality factor (Q). For critical designs use this calculator to get the starting point, then refine with a finite-element or lumped-model simulation.
Relative permeability of common core materials
| Core material | Typical mu_r | Typical application |
|---|---|---|
| Air / vacuum | 1 | RF inductors, precision coils |
| Powdered iron (Type 2) | 10 | Switching power supplies |
| Powdered iron (Type 52) | 75 | Low-loss power inductors |
| Ferrite (Ni-Zn, high freq) | 10-400 | RF, EMI suppression |
| Ferrite (Mn-Zn, low freq) | 1000-5000 | Audio, power filters |
| Silicon steel (electrical) | 1000-5000 | Transformers, motors |
| Permalloy (80% Ni) | ~8000 | Audio transformers, shielding |
| Mu-metal | 20000-100000 | Magnetic shielding |
Approximate values at room temperature and low flux density. Real values vary with frequency, temperature, and flux level.
Frequently asked questions
Why does inductance scale with the square of the number of turns?
Adding one more turn does two things at once: it contributes its own magnetic flux, and that flux links all N turns. Both the flux created per ampere and the number of turns it passes through grow with N, so the total flux linkage (and hence L) is proportional to N^2. In practice: 100 turns gives four times the inductance of 50 turns on the same former.
What core material should I use for my application?
For RF circuits (above 1 MHz), use a Ni-Zn ferrite or air core to keep losses low. For switching power supplies (10-500 kHz), use powdered iron or Mn-Zn ferrite. For 50/60 Hz mains transformers or audio work, use silicon-steel laminations. Air cores give the most stable and predictable inductance but require far more turns for the same inductance value. The reference table above lists typical permeability values.
How accurate is the formula for short or multi-layer coils?
The ideal formula is exact for an infinitely long, single-layer solenoid. For a real coil the error grows as the coil gets shorter relative to its diameter. A rough rule: if the winding length is at least three times the diameter, the error is below about 5%. For shorter coils apply the Nagaoka coefficient (a correction factor tabulated against the ratio l / (2r)). Multi-layer coils require a different, more complex formula and are better handled by a dedicated winding calculator.
What is the self-resonant frequency of a solenoid?
Adjacent turns act like small capacitors, giving the coil a distributed capacitance. The coil self-resonates when its inductive reactance equals the net capacitive reactance. Above this frequency the component behaves as a capacitor and is no longer useful as an inductor. For RF chokes the self-resonant frequency must be well above the operating frequency. A rough first estimate is f_SRF = 1 / (2 x pi x sqrt(L x C_distributed)), but C_distributed depends on winding geometry and must be measured or simulated.
How do I use the reverse-solve modes?
Choose the quantity you want to find in the "Solve for" selector, enter the desired inductance in the "Target inductance" field, fill in the remaining dimensions and core permeability, and the calculator rearranges the formula algebraically. For example, select "Number of turns", set a target of 1000 uH, enter your coil radius, length, and mu_r, and you will get the number of turns to wind.
Does this calculator account for magnetic saturation?
No - the formula assumes linear magnetic behaviour with constant permeability. In practice, core permeability drops as flux density approaches saturation, which reduces inductance and can damage the core. The insight panel flags when the computed B field exceeds typical saturation thresholds for common materials (0.4 T for ferrite, 1.5 T for silicon steel). Always check the core manufacturer datasheet for the actual saturation flux density at your operating temperature and frequency.