inductance of a closely wound toroidal coil

"The qualification that the coil should be closely wound on the toroid is to minimize the linkage flux around the individual turns of wire" what does it mean that "the linkage flux around the individual turns of wire"?

• Linkage or leakage? – Chu Jan 4 '18 at 12:18

The phrase is taken from David Cheng "Field and Wave Electromagnetics".

I agree that "the linkage flux" is a bit unfortunate term here, instead of leakage as already noticed. The meaning of "closely wound" is that every turn can be considered a radial plane (therefore no significant circumferential component of current).

what does it mean that "the linkage flux around the individual turns of wire"?

Normally in a well designed inductor, the flux from any single turn would ideally couple to every other turn. However, this is not something that occurs 100% on a practical coil because there is always some small amount of magnetism produced by each turn that manages to avoid passing through the common magnetic core of the coil and therefore this flux doesn't couple to other turns.

For an inductor this isn't a big deal; you just have to add a couple of more turns to reach the theoretical value you need. Normally we use the formula: -

Inductance of a coil = $A_L\times N^2$

Where $A_L$ is the core factor and N is the number of turns but, in a real situation, if pushed the formula becomes: -

Inductance of a coil = $A_L\times N^{(2-x)}$

Where $x$ reduces the relationship to slightly less than a turns-squared situation.

It is the presence of flux linkages that gives the inductance a near square value based on turns.

For a transformer this is more problematic because you have the situation where all the flux linkages produced by the primary don't 100% couple with the secondary and this leads to what is known as leakage inductance.