Flux linkage is a property of a two-terminal element. Although it is often confused with magnetic flux, the flux linkage is actually an extension rather than an equivalent of magnetic flux. Flux linkage is defined as

${\displaystyle \lambda =\int _{\,}{\mathcal {E}}\,dt}$

where ${\displaystyle {\mathcal {E}}}$ is the voltage across the device or the potential difference between the two terminals. This definition can also be written in differential form as

${\displaystyle {\mathcal {E}}={\frac {d\lambda }{dt}}}$

Faraday showed that the magnitude of the EMF generated in a conductor forming a closed loop is proportional to the rate of change of the total magnetic flux passing through the loop (Faraday's law). Thus, for a typical inductance (a coil of conducting wire), the flux linkage is equivalent to magnetic flux, which is the total magnetic field passing through the surface (i.e., normal to that surface) formed by a closed conducting loop coil, and is determined by the number of turns in the coil and the magnetic field; i.e.,

${\displaystyle \lambda =\int \limits _{S}\,{\vec {B}}\cdot d{\vec {S}}}$

where ${\displaystyle {\vec {B}}}$ is the flux density, or flux per unit area at a given point in space.

The simplest example of such a system is a single circular coil of conductive wire immersed in a magnetic field of constant magnitude at every point and with a direction perpendicular to the surface formed by the loop. In this case the flux linkage is simply the flux density passing through the loop multiplied by its surface area; i.e., ${\displaystyle \lambda =|{\vec {B}}|\,A=BA}$. If several turns of the wire are made, this becomes ${\displaystyle \lambda =NBA}$ where N is the number of turns. In this case, the surface through which the magnetic flux is passing has been increased by a factor of ${\displaystyle N}$.

Due to the equivalence of flux linkage and total magnetic flux in the case of inductance, it is popularly accepted that the flux linkage is simply an alternative term for total flux, used for convenience in engineering applications. Nevertheless, this is not true, especially for the case of memristor, which is also referred to as the fourth fundamental circuit element. For a memristor, the electric field in the element is not as negligible as for the case of inductance, so the flux linkage is no longer equivalent to magnetic flux. In addition, for a memristor, the flux linkage related energy is dissipated in the form of Joule heating, instead of being stored in magnetic field as done in the case of an inductance: there is no physical magnetic field involved as a link to anything! In conclusion, flux linkage should be interpreted as an extension rather than an equivalent to magnetic flux.

Flux linkage is commonly used to estimate the magnetic flux of an AC motor. However, errors in the voltage measurement and the resulting drift problem of the integral cause the ideal integral to often be replaced with a low-pass filter approximation.

## References

• [1] L. O. Chua, "Memristor-The Missing Circuit Element," IEEE Trans. Circuit Theory, vol. CT_18, no. 5, pp. 507–519, 1971.