Magnetic Reluctance

In the same way that a conductor resists establishment of electric current in an electric circuit in response to the application of an EMF, a magnetic material shows reluctance against the establishment of flux in the core (magnetic circuit) in response to the application of an MMF.

In a resistive electric circuit, Ohm’s law gives:

$$e=Ri$$

where $e$ is the electromotive force (voltage) and $i$ is the current.

In a magnetic circuit, the equivalent of Ohm’s law can be stated as follows:

$$\mathcal{F}=\mathcal{R}\varphi$$

where $\mathcal{F}$is the magnetomotive force, and $\mathcal{R}$ is the reluctance of the electric circuit in Ampere-turn/Weber ($A.t/\text{Wb}$).

• In non-magnetic materials, $\mathcal{R}$ is constant over the entire range of variations of $\mathcal{F}$ and $\varphi$, and therefore, $\mathcal{F}$ is proportional to $\varphi$.
• In magnetic materials, $\mathcal{R}$ is not a constant and thus, $\mathcal{F}$-$\varphi$ relation is not linear.

Like the case of resistance $R$ in electric circuits, $\mathcal{R}$ is a function of the material of the core and the geometry of the magnetic core.

In electric circuits, resistance R is related to the property of material and geometry of the resistor through the relation:

$$R=\frac{l}{\sigma A}(\Omega )$$

where:

• $R$ is the resistance in ohms
• $l$ is the length in meters
• $A$ is the cross-sectional area in ${\text{m}}^2$
• $\sigma$ is the conductivity of the material in siemens per meter ($S/m$)

Similarly, in a similar formula relates reluctance $\mathcal{R}$ to the property of material and geometry of the magnetic core:

$$\boxed{\mathcal{R}=\frac{l}{\mu A}(A.t/\text{Wb})}$$

where:

• $\mathcal{R}$ is the reluctance in $A.t/\text{Wb}$
• $l$ is the mean path length in $\text{m}$
• $A$ is the cross-sectional area in ${\text{m}}^2$
• $\mu$ is the permeability of the core in Henry/meter ($\text{H}/\text{m}$).