Inductor Design Equations Derivation

The main equations used in inductor design steps are derived below. These are

1. Area Product equation
2. Number of turns equation

Area Product (AP)

Area product of any magnetic core involves in the cross-sectional area and window areas of core

Area product, AP = A_E . A_W

Lets derive one by one.

There are two law’s are involved in this process

1. Faraday’s first law: Placing the current carrying conductor in a varying magnetic field, it produces the EMF.
2. Faraday’s second law: The induced EMF in a coil is equal to the rate of change of flux linkage to it.

Cross-sectional area (A_E):

Inductor is an electrical component and magnetic bar is a magnetic component and produces the magnetic field.

If a coil has ‘N’ number of turns around the magnetic bar, the EMF generation equation can write according to Faradays laws is as follows

From the electric field, the emf of coil i.e. voltage across an inductor is

When we equate these two equations,

and we take the flux in form of flux density,

From above two eqations,

Here, the flux density B reaches its maximum value in magnetic filed as the current in electrical field reaches its maximum. B –> B_m @ i –> I_peak

Since we need the cross sectional area,

Window Area (A_W):

Core copper area,

Current density of wire,

Actual copper area required to wind ‘N’ turns with a cross-sectional area of wire ‘a_w‘,

From above two equations,

Equating the core copper area and actual copper area required,

Since we need the window area,

From both area equations, we can derive area product equation

Note:

The core cross-sectional area is directly available from core data sheet but the window area is not directly available from datasheet, it needs to be calculated based on dimensions given in datasheet or physical measurements as given in core window area calculations

Number of turns

Based on selected core, the number of turns can be calculated using the cross-sectional area equation since we will get to know the information of core cross-sectional are from the core datasheet directly.

The above derived two equations are main in inductor design procedure.