# Magnetic Circuits

A magnetic circuit can be described as a complete closed path of any group of lines of magnetic flux. Magnetic flux is generated by permanent magnets, electromagnets or other types of magnetic materials and is described as a measure of the number of magnetic field lines that pass perpendicularly through a surface. There are a number good analogies between magnetic and electric circuits, for instance; magnetic flux is related to electrical current, reluctance is related to resistance and finally, what is known as magnetomotive force corresponds to electromotive force<ref> Dictionary.com </ref>. The use of magnetic circuits is very broad and extends to many electrical/mechanical devices such as motors and generators.

## Magnetomotive Force

Magnetic force, in general, can be thought of as the work that would be done to carry a unit magnetic pole around the entire magnetic circuit.<ref>Magnetic induction in iron and other metals Sir James Alfred Ewing</ref>Permanent magnets display this behavior naturally and it is constant as long as the magnet is not tampered with. In contrast, the magnetic force in electromagnets is primarily influenced by both the amount of current and the number of turns around a given core.<ref>The beginner's handbook of amateur radio Sir James Alfred Ewing]</ref>

By definition, the Magnetomotive force is found by multiplying the current I by the number of turns N in a coil, thus magnetomotive force $\mathcal{F}$is:

$\mathcal{F} = N I$

and has units of ampere-turn (At).

Magnetomotive force is also dependent on a given cores' permeability, in that a core whose permeability is very low, will not be able to carry the force very well.

### Permeability

Magnetic permeability is a measure of a materials ability to propagate magnetic flux. A higher permeability leads to a stronger magnet. The idea of permeability is similar to that of conduction. Since materials with a high conductivity allow electric current to flow easily, likewise, materials whose permeability is high, allow magnetic flux to move easier.<ref>Magnetic properties of materials workshop</ref>

The permeability of a certain material is not necessarily either constant nor linear since, by definition, a materials' permeability is based both in it's B (flux density) and H (magnetizing force) fields by:

$\mu = \frac{\mathbf{B}}{\mathbf{H}}$

Thus there are many materials characterized by their permeability values such as paramagnetic materials, whose permeability values generally are below 1 (with respect to $\mu_0$), paramagnetic materials, whose permeability values fall between 1 and 10 and finally ferromagnetic materials, whose permeability values can greatly exceed 10 and are typically non-linear. <ref>GeoPhysics </ref>Posted permeability values are typically obtained by taking the maximum value of the slope ( $\mu$ ) of a straight line from the origin that is tangent to the B/H curve, ass seen in the figure. <ref>Permeability </ref>

The permeability of a material is often measured relative to the permeability of a vacuum (also known as the permeability of free space) whose constant is $\mu_0 = 4 \pi \times 10^{-7}$ giving a relative permeability found by:<ref>Simple Nature Benjamin Crowell</ref>

$\mu_{r} = \frac{\mu}{\mu_{0}}$ Thus, materials can be defined by their deviation from 1 (as described above)

#### Permeability values for some common materials

Values obtained from Wikipedia<ref>Wikipedia - Permeability</ref>
Medium Permeability Î¼ [H/m] Relative Permeability Î¼/Î¼0
Mu-metal 25,000Ã—10-6 20,000<ref name="hyper">"Relative Permeability", Hyperphysics</ref>
Permalloy 10,000Ã—10-6 8000<ref name="hyper"/>
Electrical steel 5000Ã—10-6 4000<ref name="hyper"/>
Ferrite (nickel zinc) 20Ã—10-6-800Ã—10-6 16-640
Ferrite (manganese zinc) >800Ã—10-6 >640
Steel 875Ã—10-6 100<ref name="hyper" />
Nickel 125Ã—10-6 100<ref name="hyper" />-600
Platinum 1.2569701Ã—10-6 1.000265
Aluminum 1.2566650Ã—10-6 1.000022
Air 1.000,000,37 = 1+0.37Ã—10-6 <ref name=Cullity2008>B. D. Cullity and C. D. Graham (2008), Introduction to Magnetic Materials, 2nd edition, 568 pp., p.16</ref>
Vacuum 1.2566371Ã—10-6 (Î¼0) 1
Hydrogen 1.2566371Ã—10-6 1.0000000
Sapphire 1.2566368Ã—10-6 0.99999976
Copper 1.2566290Ã—10-6 0.999994
Water 1.2566270Ã—10-6 0.999992
Superconductors 0 0

## Reluctance

Reluctance can be defined as the opposition in a magnetic circuit to magnetic flux. It is the ratio of the magnetic potential difference to the corresponding magnetic flux.<ref>Merrian-Webster: Reluctance</ref>In a electrical circuit we have resistance, where as in a magnetic circuit, we have reluctance. Similarly where we have Ohm's law in an electrical circuit, where it states that:

$\Omega = \frac{V}{I}$

In a magnetic circuit reluctance is defined as:

$\mathcal{R} = \frac{\mathcal{F}}{\Phi}$

Where $\mathcal{R}$ is reluctance, $\mathcal{F}$ is magnetomotive force, and ${\Phi}$ is the magnetic flux. Reluctance acts the same way as resistance in a wire. Being that as the cross-sectional area of the reluctance decreases, and if the length of the magnetic material increases the reluctance increases.<ref>Sci-Toys.com: Magnetism</ref>

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