Inductor Structure - Calculating Inductance

Inductance Calculation

Inductors come in many shapes and sizes. However there are some common features in inductors.

Inductors are magnetic devices and they often have a magnetic path of ferromagnetic material (iron, nickel or cobalt).

The magnetic path is often "closed" so that the magnetic field lies entirely within the ferromagnetic material.
Some inductors will have a very small air-gap so the most of the magnetic path lies within the ferromagnetic material, but the magnetic flux has to cross the gap. That produces an inductor with more linear properties because ferromagnetic materials have nonlinear magnetic properties.
There are exceptions where the magnetic path is through air. These are called "air core" inductors.

Every inductor has wire, usually coiled around the ferromagnetic material.

By putting a current through the wire a magnetic field is produced in the ferromagnetic material.
The wire often has many "turns" as it is usually wound around and around the ferromagnetic materil

Much of this will become clearer if we consider a particular structure.

We can examine a structure often used for larger inductors. The ferromagnetic material structure is shown below in Figure 1.

Figure 1

This structure is not usually a solid piece of ferromagnetic material. It is usually built from thin laminations stacked up to give this shape. Then, given that shape, wire is wrapped around the magnetic structure. Here is an inductor with two turns of wire (shown as red lines) wrapped around the ferromagnetic structure.

Figure 2

We will want to define some terms. And, as we do that, we will refer to Figure 3.

Figure 3

The number of turns wrapped around the ferromagnetic material will be designated "N".
The cross-sectional area of the ferromagnetic material will be designated "A". The cross-sectional area is shown in Figure 3.

We can already write some of the physical relationships for the magnetic field in the material.

The voltage at the ends of the wire wrapped around the ferromagnetic material is given by:

v(t) = N df/dt
f is the magnetic flux "flowing" within the ferromagnetic material. The "right hand rule" would should the flux flowing within the magnetic material in the direction shown by the green arrow. Magnetic flux is measured in webers.

The flux, f. can be related to the magnetic field strength, B, by:

f = B*A (And, remember, A is the cross-sectional area of the ferromagnetic material. "B" is the flux density in webers/(square meter). That's all in the MKS system of units.

Ultimately we will use this relationship:

v(t) = NA dB/dt

That gives us a relationship between the voltage and the flux density in the magnetic material. But that flux density is related to the current that flows in the wire. To see that relationship we need to calculate the "magneto-motive" force that drives the magnetic field.

The driving force - commonly called the magneto-motive force, or just MMF, is given by

MMF = Ni(t)

That MMF produces a quantity H. When you get to electromagnetic fields, there is a line integral relationship that you can use. However here we will just note that:

H*length = N i(t)
length = mean magntic path length - shown in Figure 4 as a closed green line inside the ferromagnetic material. The mean magnetic path length is the total length of the green line.
So, we have H = N i(t)/length.

Figure 4

Finally, we know that B and H are related by:

B = mH
The quantity, m, is known as the permeability, and is really a product:

m = m_rm_o
And,

m_o = the permeability of free space
m_r = the relative permeability of the material - a quantity greater than 1.0
The relative permeability of a material can be large, over 1000 in many cases.

The permeability of a material may not be a simple multiplicative factor. Many magnetic materials have highly nonlinear relations between B and H. In fact, magnetic materials have a memory, and the nonlinear relations is multi-valued, forming something called a "hysterisis loop".

Now, we can put all of this together.

v(t) = NA dB/dt
B = mH

This implies v(t) = mNA [dH/dt]

H = Ni(t)/Length

And, that in turn implies:
v(t) = [mN²A/Length] [di(t)/dt]

And, by rememberingthat we have v = L di(t)/dt for an inductor, we get

L = mN²A/Length

Reflecting on this, the inductance, L, is a produce of geometric factors and the permeability. Further reflection would tell you that you can compute the permeability if you know the geometric factors and you can measure the inductance.