# Magnetic circuits

*Last Modification: May 16, 2013*

This article describes the physical characteristics of magnetic circuits and gives a summary of elementary calculations about magnetic systems. They can be used to convert magnetic parameters from measurements.

## Physical characteristics

The physical characteristics are very important when doing calculations on magnetic circuits. Essentially are the cross-sectional area of the core *A _{c}*, and the magnetic path length

*l*. An additional dimension, that's not required to determination the magnetic properties, but is necessary for the calculation of the actual inductor or transformer, is the cross-sectional coiling area

_{c}*A*.

_{w}The core-area as well as the coiling area can easily been measured with a vernier gauge. The magnetic length will have to be estimated. This is the average path that the magnetic field follows, roughly though the center of the core.

### Air-gap

In figure 2 are two options shown how an air-gap can be created in an E-core.

On the left side a standard E-core where are spacers sheets being used to create the air-gap. In this case the air-gap *l _{gap}* is equal to twice the spacer thickness. Because a complete loop of the magnetic field lines must cross twice the separations between the core halves.

On the right side a core where the air-gap is created by special core halves with a shortened middle leg. The outer legs of the core halves lie against each other without spacing. The field lines cross now only ones the air-gap, so the length of separation is equal to the length of the air-gap

*l*.

_{gap}It's very difficult to determine the area of the air-gap *A _{gap}*. The field lines in the air-gap are not completely homogeneous. The area will get bigger as the length of the air-gap is increased. If the ratio between the air-gap length and the width of the legs is very small, then it's safe to set the air-gap area equal to the core area

*A*.

_{c}## Old units

American manufacturers are still using the CGS units. The most important magnetism units with their conversion factor to SI units *(Dutch)* are listed in the table below.

quantity | CGS-name | CGS unit | conversion factor |

Magnetic field strength | Oersterd | Oe | := 1000/(4ยทπ) ≈ 79,5775 A/m |

Magnetomotive force | Gilbert | Gb | := 10/(4*&pi:) ≈ 0,7958 A |

Magnetic flux | Maxwell | Mx | := 10^{-8} Wb |

Magnetic inductance (flux density) | Gauss | Gs | := 10^{-4} T |

The CGS system of units doesn't know a physical constant for permeability. The CGS permeability is equal to the relative permeability within the SI-system.

## Elementary equations

The magnetic flux through an area *A*, for a homogeneous field:

[equ. 1]

Magnetic induction caused by a magnetic field *H* in a material:

[equ. 2]

Magnetic field strength in a closed magnetic circuit with a length *l* caused by a current *I* in a number of turns:

[equ. 3]

Voltage generated in a number of turns *N* caused by a flux change *dΦ/dt*. (Faraday's law of induction, negative sign according to Lenz's law):

[equ. 4]

Voltage generated in a self-induction *L* caused by a current change *dI/dt*:

[equ. 5]

The self-induction calculated by the number of turns *N* and the magnetic resistance of the circuit *Rm*:

[equ. 6]

Most of the times the AL-value is specified for a core in stead of the magnetic resistance:

[equ. 7]

The total magnetic resistance will be calculated from all the separate parts of the magnetic circuit, each with a length *l*, area *A* and permeability *μ*:

[equ. 8]

The separate parts consist most of the time only by the core * _{c}* and the air-gap

*:*

_{gap}[equ. 9]

The magnetic resistance can be derived from the magnetomotive force *F* and the magnetic flux *Φ*. (Hopkinson's law):

[equ. 10]

The magnetomotive force is caused by a certain current *I* through the windings of a coil:

[equ. 11]

The energy stored in a coil as function of the magnetic *Φ* and the magnetomotive force *F*:

[equ. 12]

The energy stored in a core per volume unit due to the magnetic induction *B* and the magnetic field strength *H*:

[equ. 13]