Basic and Useful Equations of Magnetic Theory

1. Faraday’s Law of Induction

The induced voltage V in a coil with N turns is equal to the rate of change of the flux F linking the N turns.

2. Faraday’s Law Applied to Rectangular Pulses

The peak to peak flux density DB relates to the effective cross sectional area of the core Ae when the voltage V is constant for the applied time ton.

3. Faraday’s Law Applied to Periodic Square Waves

The peak value of the flux density for a square wave symmetrical about the zero axis with a period equal to 2 X ton is

4. Faraday’s Law Applied with Sinusoidal Excitation

The peak value of the flux density for a sine wave applied voltage is

5. Ampere’s Law

Around a closed loop of length l of the dot product of the magnetic field strength H and incremental loop length dl is equal to the ampere-turns NI enclosed by the loop.

6. Ampere’s Law Applied to Most Core Geometries

Magnetic Path Length le in centimeters relates to the magnetic field strength H by

7. B versus H

The relationship between field strength and flux density is

is relative permeability relative to the permeability of air (H/m)

8. Lenz’z Law

The voltage current (also known as VI relationship) relationship in an inductive element is given by

9. Definition of Inductance

Inductance is equal to the flux linkages (NF) divided by the current producing the flux

10. Skin Depth

As voltages and currents are switched at higher frequencies, instead of being evenly distributed throughout the conductor, electrons migrate and are forced to flow at the surface. This change in penetration depth of the electrons in the conductor is characterized by the skin depth equation.

For copper conductors, this expression reduces in centimeters to:

11. Resistivity of Copper

The resistivity expression relates the resistance of a material to the material temperature (in C)

12. Temperature Coefficient of Resistivity per Degrees Celsius

The resistivity, and thus resistance, of metals increases with temperature displaying a positive temperature coefficient. Some commonly used metals in magnetics and their associated coefficients are:

Copper 0.0042

Aluminum 0.0042

Iron 0.006

Gold 0.0035

Tin 0.00045

For copper, the resistance of a conductor will increase by 30% when the temperature of that conductor is raised from 25 C to 100 C.

0.004(100-25) = 0.3 = 30%