Inductance of Straight Wire & Coils

Both straight wires and coils have inductance which can be calculated using recognised formulas or equations.

Inductance and Transformer Tutorial Includes:
Inductance Symbols Lenz's law Self inductance Inductive reactance calculations Inductive reactance theory Inductance of wire & coils Mutual inductance Transformers

Although coils normally come to mind when thinking of inductance, a straight wire also has inductance associated with it.

For most low frequency applications and normal wire lengths, this inductance can be ignored, but as frequencies rise the inductance of wires can become very significant.

Inductance can be calculated for both a straight wire and also for coils, although it can be more difficult to calculate the inductance of a cols as there are very many variables involved including the dimensions, constants and the like.

The advantage of using a coil where inductance is required is that it provides a very much higher level of inductance within a given volume as there is magnetic interaction between the different turns of the coil.

Calculating inductance

In general it is possible to calculate inductance from a knowledge of Maxwell's equations. However the mathematics involved may not always be easy. In addition to this high frequency signals mean that aspects like skin effect need to be accommodated as it affects issues like the surface current densities and magnetic field and these may involve the use of the Laplace equation.

Accordingly it is possible to apply some realistic simplifications to provide more usable calculations and equations for determining the inductance.

For example it can often be assumed that where "thin" wires are used, the current distribution cross the wire will be constant across the diameter of the wire, and this on its own allows for considerable simplifications in calculating the inductance of a wire.

Inductance of a single straight wire

For most applications the inductance of a straight wire is ignored. The inductance is very low and for most applications it is too small to have any significant effect on the circuit.

However as frequencies rise into the microwave region, even the inductance of short lengths of wire can have a significant effect.

Although calculating the inductance of a length of wire may appear to be a simple calculation, it is not quite as simple as it might appear at first sight. The flux of a length of wire will interact with the flux from other wires, or even the length of conductor that is connected to the length being considered.

It is possible to calculate the theoretical inductance of the internal plus external inductance of a straight length of wire at low frequencies.

L_{dc} = 2 l (\log_{e} (2 \frac{l}{r}) - 0.75)

Where:
L_dc = low frequency inductance in nanohenries
l = length of wire in cm
r = radius of the wire in cm

This calculation assumes that the radius of the wire is very much less than the length.

For high frequencies the skin effect means that the internal inductance tends to zero and the overall high frequency inductance formula becomes:

L_{hf} = 2 l (\log_{e} (2 \frac{l}{r}) - 1.0)

In view of all the interactions within a circuit, these formulae only give an approximate value for the inductance. They give a very good indication of the magnitude, but they cannot take into account al the effects oft he circuit and the external magnetic flux couple, etc.