It is possible to relate the coupling coefficient \(k\) to both the total (\(\phi_{1}\) or \(\phi_{2}\)) and shared flux (\(\phi_{12}\) or \(\phi_{21}\)) or to the self (\(L_{1}\) and \(L_{2}\)) and mutual (\(M\)) inductances: \[ k = \frac{M}{\sqrt{L_{1} L_{2}}} = \frac{\phi_{12}}{\phi_{1}} = \frac{\phi_{21}}{\phi_{2}} \]
\[ \begin{aligned} L_{1} &= \frac{N_{1} \phi_{1}}{i_{1}} & L_{2} &= \frac{N_{2} \phi_{2}}{i_{2}} \\ M &= \frac{N_{2} \phi_{12}}{i_{1}} & M &= \frac{N_{1} \phi_{21}}{i_{2}} \\ \end{aligned} \] \[ M M = \frac{N_{1} \phi_{21} N_{2} \phi_{12}}{i_{1} i_{2}} \] If \(k = \phi_{12}/\phi_{1}\) and \(k = \phi_{21}/\phi_{2}\) \[ M^{2} = \frac{N_{1} k \phi_{2} N_{2} k \phi_{1}}{i_{1} i_{2}} \] or \[ M^{2} = k^{2} \frac{N_{1} \phi_{1}}{i_{1}}\frac{N_{2}\phi_{2}}{i_{2}} = k^{2} L_{1} L_{2} \] therefore \[ k = \frac{M}{\sqrt{L_{1} L_{2}}} = \frac{\phi_{12}}{\phi_{1}} = \frac{\phi_{21}}{\phi_{2}} \]
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