calculator:solenoid_single_layer_circular_inductance
This is an old revision of the document!
Calculator of inductance of a single-layer thin round solenoid
 | Stan Zurek, Calculator of inductance of a single-layer thin round solenoid, Encyclopedia Magnetica, https://www.e-magnetica.pl/doku.php/calculator/solenoid_single_layer_circular_inductance, {updated: 2026/02/10 23:46} |
 See more: Calculators of inductance |
|
Toroidal coil dimensions: a - mean diameter (to the centre of the wire), c - wire diameter, b - solenoid length (including insulation of the wire), N - number of turns
S. Zurek, E-Magnetica.pl, CC-BY-4.0
Inductance of an ideal straight “thin” round solenoid (circular cross-section) can be calculated from the equations as specified below. Various equations apply for “long” and “short” solenoids, with limited range of validity, because there are no closed-form solutions to elliptic integrals (only their limited approximations).
The wire of the coil is assumed to be infinitely thin, and the current distribution is uniform (equivalent to the ideal current sheet configuration, with the skin effect ignored). The medium is assumed non-magnetic (permeability is unity, as it is for vacuum).
Illustration of equation validity with respect to the solenoid length (denoted as ratio of length/diameter). The most robust is equation L1(K2) which holds for the ratio of length/diameter from 0.075 upwards (so even for relatively very short solenoids through to infinitely long). The inductance value is normalised by multiplying it by the length, with a fixed number of turns.
S. Zurek, E-Magnetica.pl, CC-BY-4.0
Equations
| Approximate inductance of a straight “thin” solenoid (with a circular cross-section) | ||
|---|---|---|
| Source: [1] Clayton R. Paul. Inductance: Loop and Partial, Wiley-IEEE Press, 2009, New Jersey, ISBN 9780470461884 | ||
| Formula (1) simplistic, valid only for “infinitely long” solenoids [1], eq. (4.35), p. 137 | $$ L_0 = \frac{ μ_0 · π · a^2 · N^2 }{4·b} = \frac{ μ_0 · A · N^2 }{b} $$ | (H) |
| where: $μ_0$ - permeability of vacuum (H/m), $a$ - mean diameter (m) of solenoid measured to the centre of the wire, $b$ - solenoid length (m), $N$ - number of turns (unitless), $A = π · a^2/4 $ - cross-sectional area of the solenoid (m2) | ||
| Source: [2] Edward B. Rosa, Frederick W. Grover, Formulas and tables for the calculation of mutual and self-inductance, Bulletin of the Bureau of Standards, Vol. 8, No. 1 | ||
| Formula (2) for long and short solenoids (depending on K) [2], eq. (75), p. 119 | $$ L_1 = \frac{ μ_0 · π · a^2 · N^2}{4·b} · K_{1,2,3,4,5} = L_0 · K_{1,2,3,4,5} $$ | (H) |
| where: all the variables as defined above, and $K$ is a correction factor defined by several equations ($K = K_1$, or $K = K_2$, etc.), depending on the coil length: | ||
| Formula (2a) simplified, for long solenoids [2] Webster and Havelock, eq. (79), p. 121 | $$ K_1 = 1 - \frac{4}{3·π} + \frac{1}{8} · \frac{a^2}{b^2} - \frac{1}{64} · \frac{a^4}{b^4} + \frac{5}{1024} · \frac{a^6}{b^6} - \frac{35}{16384} · \frac{a^8}{b^8} + \frac{147}{131072} ·\frac{a^{10}}{b^{10}} $$ | unitless) |
| Formula (2b) complex, more precise, for long and short solenoids [2] eq. (77), p. 120 | $$ K_2 = \frac{2}{3·(1-δ)^2} · \left( 1 + \frac{8·β}{1+α} + \frac{k_p^2}{k^2} · \frac{8·γ}{1-δ} \right) - \frac{4}{3·π} · \frac{k}{k_p} $$ | unitless) |
| where for $K_2$ the variables are defined as (all unitless): | ||
| $ α = q^2 + q^6 + q^{12} $ $\quad \quad$ $ β = q^2 + 3·q^6 + 6·q^{12} $ $\quad \quad$ $ γ = q - 4·q^4 + 9·q^9 $ $\quad \quad$ $ δ = 2·q - 2·q^4 + 2·q^9 $ $ q = \frac{l}{2} + 2 · \left( \frac{l}{2} \right)^5 + 15 · \left( \frac{l}{2} \right)^9 $ $\quad \quad$ $ l = \frac{1 - \sqrt{k_p}}{1+\sqrt{k_p}} $ $\quad \quad$ $ k = \sqrt{ a^2 / (a^2 + b^2) } $ $\quad \quad$ $ k_p = \sqrt{ b^2 / (a^2 + b^2) } $ |
||
| Formula (2c) complex, more precise, for short solenoids [2] eq. (76), p. 120 | $$ K_3 = \frac{1}{3·π·\sqrt{q_1}·(1+α_s)^2 } · F - \frac{4}{3·π} · \frac{k}{k_p} $$ | unitless) |
| $$ F = 1 - \frac{k_p^2}{k^2} + \left[ \frac{k_p^2}{k^2} · \left( 1 + \frac{8·β_s}{1+α_s} \right) + \frac{8·γ_s}{1-δ_s} \right] · \frac{1}{2} · ln \frac{1}{q_s} $$ | ||
| where for $K_3$ the variables are defined as (all unitless), with the difference in definition of $l_s$ (as compared to $l$ above): | ||
| $ α_s = q_s^2 + q_s^6 + q_s^{12} $ $\quad \quad$ $ β_s = q_s^2 + 3·q_s^6 + 6·q_s^{12} $ $\quad \quad$ $ γ_s = q_s - 4·q_s^4 + 9·q_s^9 $ $\quad \quad$ $ δ_s = 2·q_s - 2·q_s^4 + 2·q_s^9 $ $ q_s = \frac{l_s}{2} + 2 · \left( \frac{l_s}{2} \right)^5 + 15 · \left( \frac{l_s}{2} \right)^9 $ $\quad \quad$ $ l_s = \frac{1 - \sqrt{k}}{1+\sqrt{k}} $ $\quad \quad$ $ k = \sqrt{ a^2 / (a^2 + b^2) } $ $\quad \quad$ $ k_p = \sqrt{ b^2 / (a^2 + b^2) } $ |
||
| Source: [3] Frederick W. Grover, Inductance Calculations: Working Formulas and Tables, ISA, New York, 1982, ISBN 0876645570 | ||
| Formula (2d) Nagaoka's coefficient (with Zurek's approximation, published only here), for long solenoids [3] Table 37, p. 146 | $$ K_{4} = -0.019754 · η^{-3} + 0.13393 · η^{-2} - 0.42569 · η^{-1} + 1 $$ | (unitless) |
| where: $η = b / a$ - ratio of solenoid length $b$ to its diameter $a$ (unitless) | ||
| Formula (2e) Nagaoka's coefficient (with Zurek's improvement, published only here), for short solenoids [3] eq. (119), p. 143 (also Table 36, p. 144) | $$ K_{5} = \frac{2·η}{π}·\left[ \left( g - \frac{1}{2} \right) + \frac{η^2}{8}·\left( g + \frac{1}{8} \right) - \frac{η^4}{64}·\left( g - \frac{2}{3} \right) + \frac{5·η^6}{1024}·\left( g - \frac{109}{120} \right) \right]·Z $$ | (unitless) |
| where: $η = b / a$ - ratio of solenoid length $b$ to its diameter $a$ (unitless), $g = ln (4/η) $ (unitless), and the additional correction $Z$ (unitless): $$ Z = 1 + 0.004677 · η^2 + 0.000631 · η $$ | ||
↑
| → → → Helpful page? Support us! → → → | PayPal | ← ← ← Help us with just $0.50 per month? Come on…  ← ← ← |
calculator/solenoid_single_layer_circular_inductance.1770763618.txt.gz · Last modified: 2026/02/10 23:46 by stan_zurek