Reluctance

Prof. Dr. Thorsten Jungmann (last updated 2025-12-09)

The reluctance (German: magnetischer Widerstand, also Reluktanz) describes how strongly a material/section impedes the magnetic flux in a magnetic circuit. Geometry and permeability determine its value.

Analogy: magnetic circuit/electric circuit

Reluctance corresponds to electrical resistance in an electric circuit.

In the fundamentals of electrical engineering (EE1) only for completeness

This quantity appears in the EE1 script only for the sake of completeness, because it belongs with the components inductor and capacitor. Calculations and a deeper understanding are part of alternating current theory (EE2).

1 Quantity symbol and unit

The quantity symbol of magnetic resistance is $R_{\mathrm{m}}$.
Its unit is ‘ampere per weber’ or ‘one over henry’:

$$[R_{\mathrm{m}}]=\frac{\mathrm{A}}{\mathrm{Wb}}={\mathrm{H}}^{-1}$$

2 Practical and worked examples

In a homogeneous field, the following applies:

$$R_{\mathrm{m}}=\frac{l}{\mu A},\mu ={\mu }_0{\mu }_r$$

where:

• $l$ is the path length
• $A$ is the effective cross-section perpendicular to the flux
• $\mu$ is the permeability of the material with
  • the magnetic constant ${\mu }_0$ (permeability of vacuum)
  • the relative permeability of the material ${\mu }_r$

2.1 Core without air gap (uniform cross-section)

Given: $l=0.10\mathrm{m}$, $A=1.0\cdot 10^{-4}{\mathrm{m}}^2$, ${\mu }_r=2000$.

Reluctance:

$$R_{\mathrm{m}}=\frac{l}{{\mu }_0{\mu }_{r}A}\approx \underline{3.98\cdot 10^5\frac{\mathrm{A}}{\mathrm{Wb}}}$$

Relationship:

$$\Phi =\frac{\Theta }{R_{\mathrm{m}}},B=\frac{\Phi }{A}$$

where:

• $\Phi$ is the magnetic flux density
• $\Theta$ is the magnetomotive force
• $B$ is the magnetic flux density
• $A$ is the effective cross-section perpendicular to the flux

Example:

$$\Theta =400\mathrm{A}\Rightarrow \Phi \approx 1.01\mathrm{mWb},B\approx \underline{10.1\mathrm{mT}}$$

2.2 Influence of an air gap

Air gap $l=1.0\mathrm{mm}$ with the same cross-section $A$:
$R_{\mathrm{gap}}=\frac{g}{{\mu }_{0}A}\approx \underline{7.96\cdot 10^6\frac{\mathrm{A}}{\mathrm{Wb}}}$

Total reluctance:

$$R_{\mathrm{m},\Sigma }=R_{\mathrm{core}}+R_{\mathrm{gap}}\approx 8.36\cdot 10^6\frac{\mathrm{A}}{\mathrm{Wb}}$$

For the same flux as in 2.1 ($\Phi \approx 1.0\mathrm{mWb}$), you need:

$$\Theta =R_{\mathrm{m},\Sigma }\Phi \approx \underline{8.36\cdot 10^3\mathrm{A}}$$

With $N=100$ turns: $I=\Theta /N\approx \underline{83.6\mathrm{A}}$.

2.3 Series and parallel connection of reluctances

Series paths (same flux $\Phi$)

$$R_{\mathrm{m},\Sigma }=R_{\mathrm{m},1}+R_{\mathrm{m},2}+\dots$$

Parallel branches (same magnetomotive force $\Theta$, division of flux)

$${\Phi }_i=\frac{\Theta }{R_{\mathrm{m},i}},{\Phi }_{\Sigma }=\sum _i{\Phi }_i=\Theta \sum _i\frac{1}{R_{\mathrm{m},i}}\Rightarrow \frac{1}{R_{\mathrm{m},\mathrm{eq}}}=\sum _i\frac{1}{R_{\mathrm{m},i}}$$

2.4 Target quantity: field strength

Mean field strength along a homogeneous path:

$$H\approx \frac{\Theta }{l}$$

where:

• $H$ is the magnetic field strength

Example: $l=0.10\mathrm{m}$ and $\Theta =100\mathrm{A}:$
$H\approx \underline{1000A/m}$

3 Typical orders of magnitude

• Soft-iron core (without air gap, ${\mu }_r=10^3\text{...}10^4$): $R_{\mathrm{m}}=10^4\text{...}10^6\frac{\mathrm{A}}{\mathrm{Wb}}$
• Air gap $l=0.5\text{...}2\mathrm{mm}$, $A=10^{-4}{\mathrm{m}}^2$: $R_{\mathrm{gap}}=4\cdot 10^6\text{...}1.6\cdot 10^7\frac{\mathrm{A}}{\mathrm{Wb}}$
• Result: The air gap often dominates the total reluctance.

4 Example phrasings

• ‘The reluctance of the air gap is $8\cdot 10^6\frac{\mathrm{A}}{\mathrm{Wb}}$.’
• ‘Reluctances in series add up; the air gap determines the sum.’
• ‘For the desired flux, $\Theta =R_{\mathrm{m}}\Phi$ applies.’

5 Related quantities and relationships

• Magnetomotive force:

$$\Theta =\oint \vec{H}\cdot \mathrm{d}\vec{l},\Theta =NI$$

• Magnetic flux and flux density:

$$\Phi ={\int }_A\vec{B}\cdot \mathrm{d}\vec{A},\vec{B}=\mu \vec{H}$$

• Analogy between magnetic circuit and electric circuit:

$$\Phi =\frac{\Theta }{R_{\mathrm{m}}}\leftrightarrow I=\frac{U}{R}$$

- The [[Magnetic flux|magnetic flux]] $\Phi$ corresponds to the [[Current|electric current]] $I$.
- The [[Magnetomotive force|magnetomotive force]] $\Theta$ corresponds to the [[Voltage|electric voltage]] $U$.
- The [[Reluctance|magnetic resistance]] $R_\mathrm{m}$, a.k.a. reluctance, corresponds to the [[Resistance|electrical resistance]] $R$.

• Flux linkage and inductance (ideal homogeneous circuit):

$$\Lambda =N\Phi ,L=\frac{N^2}{R_{\mathrm{m}}}$$

6 Related components

• Inductor
• Relay and contactor
• Transformer
• Electromagnet (magnetic circuit with air gap)