2.1i.4 Mutual Inductance

Mutual Inductance for Aligned Inductors

For two aligned inductors (dots in same direction) $L_1$ and $L_2$ with a mutual inductance $M$,

\usepackage{circuitikz}
\begin{document}
  \begin{circuitikz}[american]
  	\draw (-1,0) 
  	to [short, i=$i_1$] (2, 0)
  	to [L,l_=$L_1$] (2,-3) |- (-1,-3) 
  	to [open, o-o, v^>=$v_1$] (-1,0);
  	
  	\draw (6,0)
  	to [short, i=$i_2$] (3, 0)
  	to [L,l=$L_2$] (3,-3) |- (6,-3) 
  	to [open, o-o, v=$v_2$] (6,0);
  	
  	\draw [fill=black] (1.75, -.25)node(a){} circle (2pt);
  	\draw [fill=black] (3.25, -0.25)node(b){} circle (2pt);
  	\draw [] (2.5,0) to node[pos=0.5,fill=white]{M} (2.5,0);
  \end{circuitikz}

\end{document}

The following equations hold:
$$v_1 = L_1 \frac{di_1}{dt} + M \frac{di_2}{dt}$$
$$v_2 = L_2 \frac{di_2}{dt} + M \frac{di_1}{dt}$$
If the inductors are opposing,
```tikz
\usepackage{circuitikz}
\begin{document}
  \begin{circuitikz}[american]
  	\draw (-1,0) 
  	to [short, i=$i_1$] (2, 0)
  	to [L,l_=$L_1$] (2,-3) |- (-1,-3) 
  	to [open, o-o, v^>=$v_1$] (-1,0);
  	
  	\draw (6,0)
  	to [short, i=$i_2$] (3, 0)
  	to [L,l=$L_2$] (3,-3) |- (6,-3) 
  	to [open, o-o, v=$v_2$] (6,0);
  	
  	\draw [fill=black] (1.75, -0.25)node(a){} circle (2pt);
  	\draw [fill=black] (3.25, -2.75)node(b){} circle (2pt);
  	\draw [] (2.5,0) to node[pos=0.5,fill=white]{M} (2.5,0);
  \end{circuitikz}

\end{document}

...then the following equations hold:
$$v_1 = L_1 \frac{di_1}{dt} - M \frac{di_2}{dt}$$
$$v_2 = L_2 \frac{di_2}{dt} - M \frac{di_1}{dt}$$

Proof (Understand before MT using superposition!)