# Self-inductance of toroid of rectangular cross section (N=windings?)

I have found a lot of answers about how to calculate the self-inductance of toroid of rectangular cross section, however my question says that "The winding are seen as a thin homogenuous currentlayer around the core" (excuse the translation). What does that mean for N? Does it mean N=1?

• Yes. Or put 1000 windings on it and divide the current by 1000. Same result. I can only assume this odd phrase is used to deny the presence of any skin effect. – Janka May 18 '18 at 1:36
• As @Janka is saying, you care about amps/meter and that N doesn't matter. Watch Walter Lewin at MIT on Ampere's law; somewhere near the middle (around 17 minutes and later) he gets to the point you may care about. – jonk May 18 '18 at 3:30

It's just a (confusing) way of saying 'stick to the simple calculation of flux in the core, don't worry about flux in the air'.

The inductance of a wound toroid consists of the sums of two contributions ...

a) The inductance due to the field through the core

b) The inductance due to the field in the space round the wire

The inductance due to the core dominates the total, increasingly so as the core relative permeability increases. It's easy to calculate, the magnetic path length and cross section are sell defined. This inductance is what most people would stop at when they want to calculate the inductance of a wound toroid. This inductance is the same whether the wire is wound uniformly tightly to the core (closely approximating a thin homogeneous current layer), or is wound more loosely, or in a single bunch or bunches on just part of the circumference.

The inductance due to field through the air is small, and almost totally negligible compared to the core contribution for any reasonable relative permeability. It's difficult to even estimate its contribution, calculation of it would involve 3D integrals of Biot Savart. For windings in the form of a thin homogeneous current layer, its contribution is zero, as such a toroid has no external field.

"The winding are seen as a thin homogenuous currentlayer around the core" (excuse the translation). What does that mean for N? Does it mean N=1?

Regarding the winding as a "thin homogenuous currentlayer" does not help understand what the inductance is when many turns are used. However it does help understand the H-field inside the core and whether the core is potentially saturating. Whether it's 0.1 amp and 1000 turns or 100 amps and 1 turn, the same H-field is produced.

So, amps x turns (aka ampere-turns) is referred to as the magneto motive force (MMF or sometimes $F$) and this is a useful quantity to know because: -

$\Phi = \dfrac{F}{\Re}$ i.e. total flux is MMF divided by core reluctance.

H is also MMF divided by mean core length and you can convert H to flux density (B) by knowing the permability of the core material hence, you can get to an understanding about core saturation limits for the material being used.

However, to know inductance you have to use the correct number of turns and so regarding the multitude of turns as one turn doesn't help at all.

Tehnically, $n$ windings with a current of $I'$ correspond to one winding with a current of $$I = n\cdot I'$$

In practical you get magnetic flux between your windings etc, that is not in your toroid core. This is called leakage flux.

The winding are seen as a thin homogenuous currentlayer around the core

This basically tells you that you don_t have to think about the leakage flux and other weird properties of physical coils. Just go with above definition. Under these assumptions the inductance of a toroid is quite simple to calculate.