According to Maxwell's equations', time-varying currents in wires produce electromagnetic fields in the space around the wires. These electromagnetic fields can then be captured by other wires located nearby and converted to electric power. For instance, let us consider the circuit in Fig. 1. If the two inductors are in the closed proximity of each other, they can induce voltages into each other once energized. Using Faraday's law one can show that the voltage across each inductor is equal to $$\begin{equation}v_1=L_1 \frac{di_1}{dt} \pm L_{12} \frac{di_2}{dt} \end{equation}$$ $$\begin{equation}v_2=\pm L_{21} \frac{di_1}{dt} + L_2 \frac{di_2}{dt} \end{equation}$$ where $L_1$ and $L_2$ are the (self) inductances of the two coils and $L_{12}$ and $L_{21}$ are two coupling coefficients (measured in Henries). The $L_1 \frac{di_1}{dt}$ and $L_2 \frac{di_2}{dt}$ terms in the previous equations represent the voltages induced by the current going through the inductor in the coil itself, while the $\pm L_{12} \frac{di_2}{dt}$ and $\pm L_{21} \frac{di_1}{dt}$ represent the voltages induced by the current going through one inductor in the other coil. Moreover, it can also be shown that two coupling coefficients are equal to each other, which allows us to write the previous equations using the so-called mutual inductance $M$ $$\begin{equation}M=L_{21}=L_{12}\end{equation}$$ Although we will not use it in this course, it is worthwile noting that the mutual inductance satisfies $M \le \sqrt{L_1 L_2}$.

The $\pm$ sign that appears in the previous equations depends on the exact arrangement of the windings of the two inductors (coils) with respect to each other. Therefore, it is important to indicate if the two inductors are winded in the same direction or oposite direction. A simple way to describe this, is to use the dot notation (see Fig. 2). Using the dot notation, KVL can be written for the two loops shown in Fig. 2 as follows $$\begin{equation}V_1=i_1 R_1 + L_1 \frac{di_1}{dt} + M \frac{di_2}{dt} \end{equation}$$ $$\begin{equation}V_2=i_2 R_2 + M \frac{di_2}{dt} + L_2 \frac{di_2}{dt} \end{equation}$$ In AC circuits the previous equations become $$\begin{equation}V_1=i_1 R_1 + \textcolor{blue}{j} \omega L_1 i_1 + \textcolor{blue}{j} \omega M i_2 \end{equation}$$ $$\begin{equation}V_2=i_2 R_2 + \textcolor{blue}{j} \omega M i_2 + \textcolor{blue}{j} \omega L_2 i_2 \end{equation}$$

For the circuit in Fig. 3 KVL becomes (note that the location of the dots has changed compared to the previous figure - which can happen if the coils are winded in oposite direction with respect to each other) $$\begin{equation}V_1=i_1 R_1 + L_1 \frac{di_1}{dt} - M \frac{di_2}{dt} \end{equation}$$ $$\begin{equation}V_2=i_2 R_2 - M \frac{di_1}{dt} + L_2 \frac{di_2}{dt} \end{equation}$$ In AC circuits the previous equations become $$\begin{equation}V_1=i_1 R_1 + \textcolor{blue}{j} \omega L_1 i_1 - \textcolor{blue}{j} \omega M i_2 \end{equation}$$ $$\begin{equation}V_2=i_2 R_2 - \textcolor{blue}{j} \omega M i_2 + \textcolor{blue}{j} \omega L_2 i_2 \end{equation}$$

For the circuit in Fig. 4 KVL becomes $$\begin{equation}V_1=i_1 R_1 + L_1 \frac{di_1}{dt} - M \frac{di_2}{dt} \end{equation}$$ $$\begin{equation}V_2=i_2 R_2 - M \frac{di_1}{dt} + L_2 \frac{di_2}{dt} \end{equation}$$ In AC circuits the previous equations become $$\begin{equation}V_1=i_1 R_1 + \textcolor{blue}{j} \omega L_1 i_1 - \textcolor{blue}{j} \omega M i_2 \end{equation}$$ $$\begin{equation}V_2=i_2 R_2 - \textcolor{blue}{j} \omega M i_2 + \textcolor{blue}{j} \omega L_2 i_2 \end{equation}$$

Finally, for the circuit in Fig. 5 KVL becomes $$\begin{equation}V_1=i_1 R_1 + L_1 \frac{di_1}{dt} + M \frac{di_2}{dt} \end{equation}$$ $$\begin{equation}V_2=i_2 R_2 + M \frac{di_1}{dt} + L_2 \frac{di_2}{dt} \end{equation}$$ In AC circuits the previous equations become $$\begin{equation}V_1=i_1 R_1 + \textcolor{blue}{j} \omega L_1 i_1 + \textcolor{blue}{j} \omega M i_2 \end{equation}$$ $$\begin{equation}V_2=i_2 R_2 + \textcolor{blue}{j} \omega M i_2 + \textcolor{blue}{j} \omega L_2 i_2 \end{equation}$$

The above examples, allow us to write the following rule for the sign of the voltage induced by one inductor into the second magnetically coupled inductor:

1. If a defined current enters the dotted terminal on one coil, it produces a voltage in the other coil that is possitive at the dotted terminal.
2. If a defined current enters the undotted terminal on one coil, it produces a voltage in the other coil that is possitive at the undotted terminal.

Since the voltage across magnetically coupled inductors is expressed in term of the currents in the ciruit, it is usually much easier to write the mesh equations than the nodal equations.

Although magnetically coupled inductors must be physically close to each other, they are not always represented in the proximity of each other in circuit diagrams. However, using the dot notation, we can easily identify the two inductors that are magnetically coupled with each other. For instance, consider the circuit in Fig. 6. In this circuit, KVL can be written as $$\begin{equation}-V_1 + L_1 \frac{di_1}{dt} - M \frac{di_2}{dt} + R_1(i_1-i_2)\end{equation}$$ $$\begin{equation} i_2 R_2 + V_2 + L_2 \frac{di_2}{dt} - M \frac{di_1}{dt} + R_1(i_2-i_1)\end{equation}$$ while the AC equations become $$\begin{equation}-V_1 + \textcolor{blue}{j} \omega L_1 i_1 - \textcolor{blue}{j} \omega M i_2 + R_1(i_1-i_2)\end{equation}$$ $$\begin{equation}i_2 R_2 + V_2 + \textcolor{blue}{j} \omega L_2 i_2 - \textcolor{blue}{j} \omega M i_1 + R_1(i_2-i_1)\end{equation}$$ Note that the term containing the mutual inductance has a negative sign in the previous equations because:

1. Current $i_1$ enters in the undotted terminal of $L_1$, therefore it produces a positive voltage at the undotted terminal of the second coil $L_2$. For this reason, when we write KVL for the second loop in the clockwise direction of $i_2$, the voltage induced by $L_1$ in $L_2$ is negative.
2. Similarly, current $i_2$ enters in the dotted terminal of $L_2$, therefore it produces a positive voltage at the dotted terminal of the first coil $L_1$. For this reason, when we write KVL for the first loop in the clockwise direction of $i_1$, the voltage induced by $L_2$ in $L_1$ is negative.