Inductance - Self-inductance of Simple Electrical Circuits in Air

Self-inductance of Simple Electrical Circuits in Air

The self-inductance of many types of electrical circuits can be given in closed form. Examples are listed in the table.

Inductance of simple electrical circuits in air (units are in Henries when the result is multiplied by )
Type Inductance / Comment
Single layer
solenoid		 \frac{r^{2}N^{2}}{3l}\left\{ -8w + 4\frac{\sqrt{1+m}}{m}\left( K\left( \sqrt{\frac{m}{1+m}} \right) -\left( 1-m\right) E\left( \sqrt{ \frac{m}{1+m}} \right) \right) \right\}

=\frac{r^2N^2\pi}{l}\left\{ 1-\frac{8w}{3\pi }+\sum_{n=1}^{\infty } \frac {\left( 2n\right)!^2} {n!^4 \left(n+1\right)\left(2n-1\right)2^{2n}} \left( -1\right) ^{n+1}w^{2n}\right\}
=\frac {r^2N^2\pi}{l}\left( 1 - \frac{8w}{3\pi} + \frac{w^2}{2} - \frac{w^4}{4} + \frac{5w^6}{16} - \frac{35w^8}{64} + ... \right) for w << 1
for w >> 1

 : Number of turns
r: Radius
l: Length
w = r/l

: Elliptic integrals
Coaxial cable,
high frequency		 a1: Outer radius
a: Inner radius
l: Length
Circular loop r: Loop radius
a: Wire radius
Rectangle \frac {1}{\pi}\left(b\ln{\frac {2 b}{a}} + d\ln{\frac {2d}{a}} - \left(b+d\right)\left(2-Y\right) +2\sqrt{b^2+d^2} -b\cdot\operatorname{arsinh}{\frac {b}{d}}-d\cdot\operatorname{arsinh}{\frac {d}{b}} \right) b, d: Border length
d >> a, b >> a
a: Wire radius
Pair of parallel
wires		 a: Wire radius
d: Distance, d ≥ 2a
l: Length of pair
Pair of parallel
wires, high
frequency		 a: Wire radius
d: Distance, d ≥ 2a
l: Length of pair
Wire parallel to
perfectly
conducting wall		 a: Wire radius
d: Distance, d ≥ a
l: Length
Wire parallel to
conducting wall,
high frequency		 a: Wire radius
d: Distance, d ≥ a
l: Length

The symbol μ0 denotes the magnetic constant (4π×10−7 H/m). For high frequencies the electric current flows in the conductor surface (skin effect), and depending on the geometry it sometimes is necessary to distinguish low and high frequency inductances. This is the purpose of the constant Y: Y = 0 when the current is uniformly distributed over the surface of the wire (skin effect), Y = 1/4 when the current is uniformly distributed over the cross section of the wire. In the high frequency case, if conductors approach each other, an additional screening current flows in their surface, and expressions containing Y become invalid.

Read more about this topic:  Inductance

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