# Magnetic circuits analysis

In this post, are shown the necessary concepts to make analysis of magnetic circuits, which is very similar to that of electric circuits.

## Why analyze magnetic circuits?

The knowledge about analysis of this type of circuit is useful for design components that use magnetic field, as for exemple: relays, transformers, electric motors, generators, speakers, etc.

## Magnetic field and magnetic permeability

Magnetic density flux ($B$) is defined by magnetic flux ($\phi$), in $Wb$ (Weber), divided by area ($A$), in $m^{2}$.

$B=\frac{\phi}{A}$

The measurement unit for $B$ is $Wb/m^{2}$ or $T$ (Tesla). The following equation is the relation between flux density and magnetizing force $H$, the latter is in $Ae/m$ (ampère-turn per meter).

$B=\mu H$

Where $\mu$ is the magnetic permeability. This is the product of vacuum’s magnetic permeability ($\mu _{o}$), a constant whose value is $4\pi \cdot 10^{-7}H/m$, with relative permeability ($\mu _{r}$), which depends on material.

$\mu =\mu _{o}\mu _{r}$

## Analogy with electric circuits

### Reluctance

As shown in the post about resistance, capacitance, inductance, impedance and reactance, the material’s resistance is calculated using the formula:

$R=\rho \frac{L}{A}$

The materials have reluctance ($\Re$), which is a material resistance to magnetic flux, whose equation is:

$\Re=\frac{l}{\mu A}$

Where $l$ is material’s length and $A$ is cross section’s area. The reluctance is measured in rels or $Ae/Wb$ (ampère-turn per weber).

### Ohm’s law for magnetic circuits

In a magnetic circuit, flux generators are coils, which produce the magnetomotive force (mmf) ($\mathfrak{F}$), in ampère-turn, that is coil’s current I multiplied by the number of turns N.

$\mathfrak{F}=NI$

The Ohm’s law equation for magnetic circuits.

$\mathfrak{F}=\phi \cdot \Re$

### Ampère’s circuit law for magnetic circuits

Kirchhoff’s voltage law says that the sum of voltages and voltage drop in a closed loop circuit is zero. As shown in the post, whose link is below.

Ampère’s circuit law claims that the algebraic sum of elevations and drops of magnetomotive force in a closed loop will always be zero.

$\sum \mathfrak{F}=0$

### Magnetic flux

Just like in Kirchhoff’s current law, algebraic sum of fluxes entering in a node is equal to sum of fluxes going out of this node.

## Air gap

It’s the air space between ferromagnetic materials. Since air’s relative magnetic permeability is only 1, the magnetizing force H and mmf are higher on air gap than on core.

Obtaining the equation to the magnetic circuit above, using Ampère’s circuit law.

$NI=H_{c}\cdot l_{c}+H_{g}\cdot l_{g}$

• $H_{c}$ and $H_{g}$ are magnetizing forces of core and air gap, respectively.