Helical Coil Calculator
Enter your coil geometry to calculate inductance (Wheeler formula), coil height, total wire length, wire volume, resistance, magnetic field strength, stored energy, and resonant frequency. Switch between metric and imperial units at any time. The "show your work" panel walks through every formula with your actual numbers.
Formula
Worked example
A copper coil with 30 mm diameter, 1 mm wire, 20 turns, 0.5 mm spacing: Dc = 1.181 in, pitch = 0.0591 in, Lw = 76.5 in. L = (1.181 x 400) / (18 x 1.181 + 40 x 76.5) = 472.4 / 3081.2 = 0.153 uH.
What is a helical coil?
A helical coil is a conductor wound in a helix along a cylindrical axis. When current flows through it, the coil acts as an inductor: it stores energy in a magnetic field and opposes changes in current. The simplest and most common form is the single-layer air-core coil used throughout RF electronics, antenna matching networks, Tesla coils, and power supplies. The key geometric parameters are the coil diameter, the wire diameter, the number of turns, and the spacing between adjacent turns.
Wheeler formula for inductance
The inductance of a single-layer air-core coil is most commonly calculated with the Wheeler (1928) formula: L = (Dc * N^2) / (18 * Dc + 40 * Lw), where Dc is the coil diameter in inches, N is the number of turns, and Lw is the total wire length in inches. The result is in microhenries. Wheeler derived this empirical formula by curve-fitting to measured data; it is accurate to within about 1 % for coils where the length-to-diameter ratio is 0.4 or more. For shorter coils (length much smaller than diameter), Nagaoka coefficient corrections improve accuracy.
Wire length, height, and resistance
The axial height of the coil is H = N x (Dw + S), where Dw is the wire diameter and S is the gap between adjacent turns. The wire follows a helical path, so its true length per turn is sqrt((pi x Dc)^2 + (Dw + S)^2), not just pi x Dc. The extra term accounts for the axial advance per turn; it matters most for tightly spaced or thick-wire coils. The DC resistance is R = rho x Lw / A, where rho is the conductor resistivity in ohm-metres and A = pi x (Dw/2)^2 is the cross-sectional area. Resistance affects the Q-factor (Q = 2 x pi x f x L / R) and the L/R time constant.
Magnetic field, stored energy, and self-resonance
The axial magnetic field at the centre of the coil (solenoid approximation) is H = N x I / l, where l is the coil height and I is the current. The magnetic flux density is B = mu0 x H, where mu0 = 4 x pi x 10^-7 H/m. Energy stored in the inductor is E = 0.5 x L x I^2. The L/R time constant (tau = L/R) describes how quickly current builds up or decays. Every real coil also has a distributed self-capacitance C that forms a parallel LC resonator with the inductance; the self-resonant frequency is f = 1 / (2 x pi x sqrt(L x C)). Above this frequency, the coil behaves as a capacitor rather than an inductor.
Typical inductance ranges by application
| Application | Typical inductance range | Frequency range |
|---|---|---|
| VHF/UHF RF circuits | 1 - 100 nH | 100 MHz - 3 GHz |
| HF/shortwave radio (RF coils) | 0.1 - 10 uH | 3 - 30 MHz |
| Medium-wave AM radio | 50 - 500 uH | 500 kHz - 2 MHz |
| Tesla coil secondary | 10 - 100 mH | 50 - 500 kHz |
| Audio crossover filters | 0.1 - 10 mH | 20 Hz - 20 kHz |
| Power supply chokes | 10 uH - 10 mH | 10 kHz - 1 MHz |
Air-core single-layer helical coils. Values are approximate; actual results depend on geometry.
Frequently asked questions
What is the Wheeler formula and when is it accurate?
The Wheeler (1928) formula calculates the inductance of a single-layer air-core coil as L (uH) = (Dc x N^2) / (18 x Dc + 40 x Lw), with Dc and Lw in inches. It is accurate to about 1 % for coils where the winding length is at least 40 % of the coil diameter. For pancake-style coils (length much less than diameter), accuracy degrades and Nagaoka or modified Wheeler corrections should be used.
How does the number of turns affect inductance?
Inductance scales roughly with N^2 for a fixed coil size, because both the number of flux linkages per ampere and the total flux increase with N. However, the Wheeler formula includes Lw (which also increases with N) in the denominator, so the actual scaling is slightly less than quadratic. To double the inductance, multiply turns by approximately sqrt(2) while keeping the geometry fixed.
What is the self-resonant frequency and why does it matter?
Every coil has distributed capacitance between adjacent turns. This parasitic capacitance (C) resonates with the inductance (L) at the self-resonant frequency (SRF) given by f = 1 / (2 x pi x sqrt(L x C)). Below the SRF the coil behaves as an inductor; above it the capacitive impedance dominates. For RF applications, you must choose a coil whose SRF is well above your operating frequency, typically by a factor of 3 to 10 or more.
How do I reduce the DC resistance of my coil?
DC resistance depends on wire material (copper is standard), wire diameter (larger diameter means lower resistance), and wire length. The most effective single change is to use thicker wire. A doubling of wire diameter reduces resistance by a factor of 4 (area scales as diameter squared). Switching from copper to silver reduces resistance by about 8 %, barely worth the cost except at very high frequencies. Litz wire can reduce AC resistance at high frequencies by mitigating skin and proximity effects.
What is the L/R time constant?
The time constant tau = L/R is the time it takes the current in the coil to reach about 63 % of its final value when a step voltage is applied (or to fall to about 37 % when the voltage is removed). It is the inductive equivalent of the RC time constant in capacitor circuits. A smaller resistance (better wire or thicker gauge) lengthens the time constant, while a smaller inductance shortens it.
How does spacing between turns change inductance?
Increasing the gap (S) between turns increases the coil height while keeping the diameter and number of turns fixed. A taller, more spread-out coil has a longer wire length (Lw), which enters the denominator of the Wheeler formula and reduces inductance. Close-wound coils (S = 0) give the highest inductance for a given number of turns and diameter, but also the highest self-capacitance due to closer turn proximity.