Magnetomotive force

Prof. Dr. Thorsten Jungmann (last updated 2025-09-04)

The magnetomotive force (German: magnetische Durchflutung, also magnetomotorische Kraft) is the driving cause of a magnetic field along a closed path. It sums the magnetic field strength along the path and, in a magnetic circuit, is the quantity that ‘drives’ the magnetic flux.

Analogy: magnetic circuit/electric circuit

The magnetomotive force in a magnetic circuit corresponds to the voltage in an electric circuit. This is why the magnetomotive force is also called magnetic tension (German: magnetische Spannung).

Vector quantity in a homogeneous field → scalar quantity

This is a vector quantity. In this script, however, it is treated by convention as a scalar quantity in the context of the homogeneous field (for a detailed justification).

1 Quantity symbol and unit

The quantity symbol of the magnetomotive force is the Greek capital letter $\Theta$ (‘Theta’).
Its unit is the ampere with the unit symbol $\mathrm{A}$.

$$[\Theta ]=\mathrm{A}$$

2 Practical and worked examples

2.1 Basic equations

$$\Theta =\oint \vec{H}\cdot \mathrm{d}\vec{l}=N\cdot I$$

where:

• $H$ the magnetic field strength
• $l$ the mean length of the magnetic flux path
• $N$ the number of turns
• $I$ the current in the inductor

2.2 Magnetic circuit analogies

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

where:

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

2.3 Inductor with number of turns and current

$$\Theta =N\cdot I$$

Example: $N=200$, $I=0.50\mathrm{A}\Rightarrow \Theta =100\mathrm{A}$.
Mean field strength on a closed path of length $l=0.10\mathrm{m}$:
$H\approx \frac{\Theta }{l}=\frac{100}{0.10}=1000A/m$

Simple magnetic circuit without air gap (uniform cross-section)

$$\Theta =R_{\mathrm{m}}\Phi ,R_{\mathrm{m}}=\frac{l}{{\mu }_0{\mu }_{r}A}$$

Example: $l=0.10\mathrm{m}$, $A=1.0\cdot 10^{-4}{\mathrm{m}}^2$, ${\mu }_r=2000$, desired flux $\Phi =1.0\mathrm{mWb}$.
$\Rightarrow R_{\mathrm{m}}\approx 3.98\cdot 10^5\frac{\mathrm{A}}{\mathrm{Wb}}$, $\Theta \approx 3.98\cdot 10^2\mathrm{A}$.
With $N=100$ turns: $I\approx 4.0\mathrm{A}$.

Effect of an air gap

$$R_{\mathrm{gap}}=\frac{l}{{\mu }_{0}A},\Theta =(R_{\mathrm{core}}+R_{\mathrm{gap}})\Phi$$

Example as above with an additional air gap $l=1.0\mathrm{mm}$:
$R_{\mathrm{gap}}\approx 7.96\cdot 10^6\frac{\mathrm{A}}{\mathrm{Wb}}$ dominates,
$\Theta \approx 8.36\cdot 10^3\mathrm{A}$.
With $N=100$: inductor current $I\approx 83.6\mathrm{A}$.

3 Typical orders of magnitude

• Small relays, magnet systems with soft iron: $\Theta \approx 10\dots 200\mathrm{A}$
• Small electromagnets/lifting magnets: $\Theta \approx 10^2\dots 10^3\mathrm{A}$
• Stronger magnet coils (solenoids), laboratory electromagnets (with air gap): $\Theta \approx 10^3\dots 10^5\mathrm{A}$
  (large air gap → considerably higher magnetomotive force required)

4 Example phrasings

• ‘The magnetomotive force is $250\mathrm{A}$ (amperes).’
• ‘For the required flux, $\Theta \approx 8.3\mathrm{kA}$ is needed; the air gap dominates the reluctance.’
• ‘In the magnetic circuit, $\Phi =\Theta /R_{\mathrm{m}}$ applies.’

5 Related quantities and relationships

• Magnetic field strength and line integral:
  $\Theta =\oint \vec{H}\cdot \mathrm{d}\vec{l}$
• Magnetic flux density and flux:
  $\vec{B}=\mu \vec{H},\Phi ={\int }_A\vec{B}\cdot \mathrm{d}\vec{A}$
• Reluctance model:
  $R_{\mathrm{m}}=\frac{l}{\mu A},\Phi =\frac{\Theta }{R_{\mathrm{m}}}$
• Flux linkage and inductance:
  $\Lambda =N\Phi ,\Lambda =LI\Rightarrow L=\frac{N^2}{R_{\mathrm{m}}}(\text{ideal, homogeneous magnetic circuit})$

6 Related components

• Inductor
• Relay and contactor
• Transformer