Kirchhoff’s voltage law (KVL), also known as Kirchhoff’s second law or Kirchhoff’s loop rule, states that for any closed loop in a circuit, the sum of the potential differences (voltages) across all components is zero: $$\begin{equation}\sum_{i=1}^{n} V_i = 0\end{equation}$$ where $n$ is the total number of voltages measured. Notice that, in the case of time-dependent circuits, Kirchhoff’s voltage law holds at any moment in time and can be written explicitly as $\sum_{i=1}^{n} V_i(t) = 0$.

For instance, consider the circuit shown in Fig. 1. If we go clockwise and add up the voltages across each component, KVL can be written as $$\begin{equation}R_1 I_0+V_2+R_2 I_0+V_3+R_5 I_0+R_4 I_0 + R_3 I_0-V_1=0\end{equation}$$ where we took the voltage of each source with positive sign if we went from the positive to the negative electrode and have applied Ohm's law when computing the voltage across each resistor. Notice that we can also write KVL by adding up the voltages going counterclockwise, in which case we have $$\begin{equation}-R_1 I_0+V_1-R_3 I_0-R_4 I_0-R_5 I_0-V_3 - R_2 I_0-V_2=0\end{equation}$$ however, the last two equations are identical if we we move the terms to the right hand side change the order in which they appear.

If $I_0$ is taken in opposite direction (see Fig. 2), and we write KVL counter-clockwise, we have $$\begin{equation}R_1 I_0+V_1+R_3 I_0+R_4 I_0+R_5 I_0-V_3 + R_2 I_0-V_2=0\end{equation}$$