Electronics/Inductors/Special Cases

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This section list formulas for inductances in specific situations. Beware that some of the equations are in Imperial units.

The permeability of free space, μ₀, is constant and is defined to be exactly equal to 4π×10^-7 H m^-1.

1. Basic inductance formula for a cylindrical coil

L={\frac {\mu _{0}\mu _{r}N^{2}A}{l}}

L = inductance / H
μ_r = relative permeability of core material
N = number of turns
A = area of cross-section of the coil / m²
l = length of coil / m

2. The self-inductance of a straight, round wire in free space

L_{self}={\frac {\mu _{0}b}{2\pi }}\left[\ln \left({\frac {b}{a}}+{\sqrt {1+{\frac {b^{2}}{a^{2}}}}}\right)-{\sqrt {1+{\frac {a^{2}}{b^{2}}}}}+{\frac {a}{b}}+{\frac {\mu _{r}}{4}}\right]

L_self = self inductance / H
b = wire length /m
a = wire radius /m
$\mu _{r}$ = relative permeability of wire

If you make the assumption that b >> a and that the wire is nonmagnetic ( $\mu _{r}=1$ ), then this equation can be approximated to

L_{self}={\frac {\mu _{0}b}{2\pi }}\left[\ln \left({\frac {2b}{a}}\right)-3/4\right]

(for low frequencies)

L_{self}={\frac {\mu _{0}b}{2\pi }}\left[\ln \left({\frac {2b}{a}}\right)-1\right]

(for high frequencies due to the skin effect)

L = inductance / H
b = wire length / m
a = wire radius / m

The inductance of a straight wire is usually so small that it is neglected in most practical problems. If the problem deals with very high frequencies (f > 20 GHz), the calculation may become necessary. For the rest of this book, we will assume that this self-inductance is negligible.

3. Inductance of a short air core cylindrical coil in terms of geometric parameters:

L={\frac {r^{2}N^{2}}{9r+10l}}

L = inductance in μH

r = outer radius of coil in inches

l = length of coil in inches

N = number of turns

4. Multilayer air core coil

L={\frac {0.8r^{2}N^{2}}{6r+9l+10d}}

L = inductance in μH

r = mean radius of coil in inches

l = physical length of coil winding in inches

N = number of turns

d = depth of coil in inches (i.e., outer radius minus inner radius)

5. Flat spiral air core coil

$L={\frac {r^{2}N^{2}}{(2r+2.8d)\times 10^{5}}}$

L = inductance / H

r = mean radius of coil / m

N = number of turns

d = depth of coil / m (i.e. outer radius minus inner radius)

Hence a spiral coil with 8 turns at a mean radius of 25 mm and a depth of 10 mm would have an inductance of 5.13µH.

6. Winding around a toroidal core (circular cross-section)

L=\mu _{0}\mu _{r}{\frac {N^{2}r^{2}}{D}}

L = inductance / H
μ_r = relative permeability of core material
N = number of turns
r = radius of coil winding / m
D = overall diameter of toroid / m

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