Electronics/Inductors/Special Cases

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

The permeability of free space, &mu;0, is constant and is defined to be exactly equal to 4&pi;&times;10-7 H m-1.

1. Basic inductance formula for a cylindrical coil

 * $$L=\frac{\mu_0\mu_rN^2A}{l}$$


 * L = inductance / H
 * &mu;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] $$


 * Lself = 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:

 * $$L=\frac{r^2N^2}{9r+10l}$$
 * L = inductance in &mu;H
 * r = outer radius of coil in inches
 * l = length of coil in inches
 * N = number of turns

4. Multilayer air core coil

 * $$L = \frac{0.8r^2N^2}{6r+9l+10d}$$
 * L = inductance in &mu;H
 * 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
$$L=\frac{r^2N^2}{(2r+2.8d) \times 10^5}$$
 * 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)

 * $$L=\mu_0\mu_r\frac{N^2r^2}{D}$$


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