# Toroid Core Permeability Calculation

By below mentioned calculations we can easily find unknown relative permeability of a toroid core. Which will help you to find the working frequency level of the toroid core.

### What is Relative Permeability?

Relative permeability defines that how the existence of a specific material affects the relationship between flux density and the magnetic field strength. The word 'relative' states that the permeability is defined in relation to the permeability of a vacuum.

### What is Initial Permeability?

Initial permeability defines the relative permeability of a material at low values of the Magnetic Flux Density (i.e. below 0.1 Tesla). Low flux has the benefit that every ferrite can be easily measured at that density without risk of the saturation.

Comparison of different ferrite material can be easily done by with the help of permeability.

### Permeability Calculation of Toroid Core

Step No. 1: Find the AL value with the help of test winding. Wind several turns of copper wire on the core, or if you take out the core from a circuit then use an existing winding.

Record the measured inductance value. Convert it in Nano-Henry.
Then count the number of turns i.e. N.

Then AL value can be calculated by:

AL = L/N²

The above formula can be re-arranged to calculate the required number of turns for a particular inductance as mentioned below;

N = √ (L/ AL)

It can be further re-arranged to give inductance (nH) for a particular number of turns as mentioned below;

L = AL N²

Further steps will help to identify the actual used core material; by which we can easily find the range of frequencies over which it is useful or works.

Permeability can be calculate using the method given in the below steps.

Step No. 2: Below diagram is of toroid core. First measure the toroid core dimensions in millimetre.

Where,
Inside diameter = ID
Outside diameter = OD
Height = ht.

Step No. 3: Calculate the core cross sectional area Ae and the magnetic path length Le.

Cross sectional area is given by:

Ae = [(OD-ID) h / 2]-πr²

The magnetic path length is given by:

Le = [(OD-ID) π] / 2

Step No. 4: Calculate relative permeability μr:

μr = (AL Le) / (0.4 π Ae)

By placing the formulas of Ae and Le, the formula of μr can also be written as:

μr = AL{(OD+ID) / [0.4ht(OD-ID)-0.8πr²]}

Mostly in datasheet the ferrite manufacturers provide the details of initial permeability μi, which for most practical purposes are the same as the value calculated for relative permeability μr.

Further we can also calculate the permeability of a core with air gap: effective permeability.
It can be calculated by below formula;

μe= μr/ {[(igap/icore)μr] + 1}

Where,
1 = Permeability of air gap
igap = gap length
icore = length of the core
μr = relative permeability