Kirchhoff's Laws

Kirchhoff’s Current Law

For any node in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of the node.

{% raw %} $$ \sum_{k=1}^{n} I_k = 0 $$ {% endraw %}

Kirchhoff’s Voltage Law

The directed sum of the potential differences (voltages) around any closed loop is zero.

{% raw %} $$ \sum_{k=1}^{n} V_k = 0 $$ {% endraw %}

Usage

We can use these two laws to find the current and voltage at any point in a linear circuit. To do so, we create a system of linear equations to solve.

From the above example, we get these three equations:

{% raw %} $$ i_1 - i_2 - i_3 = 0 $$

$$ - i_1R_1 - i_2R_2 + \mathcal{E}_1 = 0 $$

$$ - \mathcal{E}_1 + i_2R_2 - i_3R_3 - \mathcal{E}_2 = 0 $$

{% endraw %}

which can be solved using linear algebra:

{% raw %}

$$ \begin{bmatrix} 1 & -1 & -1 \\ -R_1 & -R_2 & 0 \\ 0 & R_2 & -R_3 \end{bmatrix} \begin{bmatrix} i_1 \\ i_2 \\ i_3 \end{bmatrix}

\begin{bmatrix} 0 \\ -\mathcal{E}_1 \\ \mathcal{E}_1 + \mathcal{E}_2 \end{bmatrix}

$$

{% endraw %}