Thévenin's theorem states that any two-terminal linear circuit containing only voltage sources, current sources and resistors (or impedances in the case of AC circuits) can be replaced by an equivalent combination of a voltage source $V_{Th}$ in series with a resistor $R_{Th}$ (or impedance $Z_{Th}$ in the case of AC circuits).

Thévenin's theorem and its dual, Norton's theorem, are widely used to simplify circuit analysis and study a circuit's initial-condition and steady-state response. Thévenin's theorem may in some cases be more convenient to use than Kirchhoff's circuit laws when analyzing linear circuits.

Calculating the Thévenin equivalent circuit, or simply the Thévenin equivalent of a circuit with two terminals means computing the Thévenin voltage and Thévenin resistance of the circuit.

The Thévenin voltage can be computed using any of the following methods:

• $V_{Th} = V_{oc}$ - the Thévenin voltage is equal to the open-circuit voltage at the output terminals of the original circuit. To compute $V_{oc}$ one can use any of the methods of linear circuit analysis such as nodal analysis, mesh analysis, or current and voltage division.
• If Norton current $I_N$ and Norton resistance $R_{N}$ are known, one can calculate the Thévenin voltage using $V_{Th} = I_{N} R_{N}$ .

The Thévenin resistance can be computed using any of the following methods:

• $R_{Th}$ can be computed by deactivating all independent sources and using the test voltage or test current methods using Ohm's law, $R_{Th}=\frac{V_{test}}{I_{test}}$. To use the test voltage method, we connect a test voltage source at the output terminals (usually, but not necessarily, taken as $V_{test}=1~V$) and compute current $I_{test}$. To use the test current method, we connect a test current source at the output terminals (usually, but not necessarily, taken as $I_{test}=1~A$) and compute voltage $V_{test}$.
• If the circuit does not contain any dependent sources, $R_{Th}$ can be computed by deactivating all independent sources and looking at the resistance seen from the output terminals (in this case, $R_{Th}$ can often be computed using series and parallel simplifications of resistors).
• If $I_N$ and $V_{Th}$ are known and $I_{N} \neq 0$, one can calculate the Thévenin resistance using $R_{Th} = \frac{V_{Th}}{I_{N}}$. However, please note that if $I_{N}=0$, we cannot divide $\frac{V_{Th}}{I_{N}}$; this does not mean that $R_{Th}$ is equal to $0$ (or $\infty$), but it only means we need to use other methods to compute $R_{Th}$.

Table 1 shows possible ways to compute $R_{Th}$ and $V_{Th}$. However, please note that these are not the only ways to compute the Thévenin components, and you can often come up with alternative ways.

¹⁾ You can use either the test voltage or test current method. They should both give you the same result.
²⁾ To deactivate the independent sources in the circuit, you need to replace all the independent voltage sources with short circuits (wires) and all the independent current sources with open circuits (remove them). Make sure you do not modify the dependent sources.

• Sometimes, the Thévenin voltage and Thévenin resistance can be computed simultaneously by performing successive source transformations and using series and parallel combinations of resistors, voltage sources and current sources.
• The Thévenin voltage of a circuit that does not contain any independent voltage and current sources is always 0.
• The Thévenin and Norton resistances, the Thévenin voltage and the Norton current satisfy the following relationships: $$\begin{equation}R_{Th}=R_N\end{equation}$$ $$\begin{equation}V_{Th}=I_N R_{Th}\end{equation}$$ Because of these relationships, it is usually necessary to find only two quantities because the other two can be calculated afterwards.
• The above techniques can also be used to compute the Thévenin voltage and Thévenin impedance $Z_{Th}$ in linear AC circuits.

• AC Norton-Thévenin equivalent circuits:
  DC circuit with 2 loops and one independent source DC circuit with 3 loops and 2 independent sources DC circuit with 3 loops and 2 sources