stylistic:style_transfer, structural:expansion, quality:correction, stylistic:tone
Coupling coefficient

The coupling coefficient is the ratio of the open-circuit actual voltage ratio to the ratio that would be obtained if all the flux coupled from one magnetic circuit to the other. The coupling coefficient is related to mutual inductance and self inductances in the following way. From the two simultaneous equations expressed in the two-port matrix the open-circuit voltage ratio is found to be:

V 2 V 1 open circuit = M L 1 {\displaystyle {V_{2} \over V_{1}}_{\text{open circuit}}={M \over L_{1}}} where

while the ratio if all the flux is coupled is the ratio of the turns, hence the ratio of the square root of the inductances

V 2 V 1 max coupling = L 2 L 1 {\displaystyle {V_{2} \over V_{1}}_{\text{max coupling}}={\sqrt {L_{2} \over L_{1}\ }}}

thus,

M = k L 1 L 2 {\displaystyle M=k{\sqrt {L_{1}\ L_{2}\ }}} where

The coupling coefficient is a convenient way to specify the relationship between a certain orientation of inductors with arbitrary inductance. Most authors define the range as 0 ≤ k < 1 {\displaystyle 0\leq k<1}, but some define it as − 1 < k < 1 {\displaystyle -1<k<1\,}. Allowing negative values of k {\displaystyle k} captures phase inversions of the coil connections and the direction of the windings.