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According to the formula \$K=M/\sqrt{L_1 L_2}\$, it seems that k is inversely proportional to the inductance values of the two coils. However, it seems to me that a coupling between two 2uH inductor should be weaker than say between a 2uH and a 2mH inductor. I also came across formulae like the following in which there seems to be no dependence on inductance value at all. Another paper also stated that coefficient of coupling is independent of number of turns. All of these conflicting things are extremely confusing to me. Can someone please clear this up?

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However, it seems to me that a coupling between two 2uH inductor should be weaker than say between a 2uH and a 2mH inductor.

The coupling is determined by the physical construction of the transformer -- in particular its core -- and reflects the ratio of the common and stray magnetic fluxes. Specifically, \$K\$ is the fraction of coupled magnetic flux and \$1-K\$ is the fraction of uncoupled magnetic flux.

This video is a good overview of the two different ways transformers are modeled:

Mutual inductance versus transformer: the power electronics perspective

https://www.youtube.com/watch?v=qLoXOlrqZCU

See slide #4 at time 1:39 for a definition of \$K\$ in terms of the common and stray magnetic fluxes.

See the next couple slides for how \$M\$ is used in simulation software to model a transformer. Basically, \$M\$ is used as a coefficient for current-controlled voltage sources in the transformer simulation.

Another good video is this one which derives the modelling of a transformer from first principles:

Demystifying the meaning of leakage inductance of transformers and coupled inductors

https://www.youtube.com/watch?v=TUfaN40T1Yg

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