Thévenin's theorem plays a pivotal role in electrical circuit analysis, offering a solution to the challenges posed by variable loads within a circuit. In practical applications, it is common to encounter circuits where certain elements remain fixed while others fluctuate, often referred to as the "load." A typical household electrical outlet serves as a prime example of a variable load, as it can be connected to a variety of appliances, each with its own unique electrical characteristics. Every time a new appliance is plugged in or removed, the entire circuit requires a fresh analysis. Thévenin's theorem provides a technique that replaces the fixed portion of the circuit with an equivalent circuit, thereby simplifying analysis and design.

Thévenin's theorem states that a linear two-terminal circuit can be represented by an equivalent circuit consisting of a voltage source in series with a resistor. This equivalent circuit effectively encapsulates the behavior of the original circuit from an external perspective. Two critical parameters define this Thévenin equivalent circuit: the Thévenin equivalent voltage (V_Th), which corresponds to the open-circuit voltage at the terminals, and the Thévenin equivalent resistance (R_Th), which represents the input or equivalent resistance at the terminals when all independent sources are deactivated.

To illustrate the concept further, consider two circuits shown in Figure 1 and Figure 2. These circuits are deemed equivalent when they share the same voltage-current relationship at their terminals. When the terminals are left open-circuited, resulting in no current flow, the voltage across the open terminals will be the Thévenin voltage. When all independent sources are turned off. Then the resistance seen across the terminals is the Thévenin resistance.

Figure 1

Figure 2

To practically apply this idea and determine the Thévenin resistance, we must consider two scenarios:

CASE 1: In situations where the network contains no dependent sources, all independent sources are deactivated and the input resistance is ascertained, which is the resistance seen between terminals a and b.

CASE 2: When the network incorporates dependent sources, continue to deactivate all independent sources. However, refrain from deactivating dependent sources since they are controlled by circuit variables.

It is worth noting that the value of Thévenin resistance (R_Th) can sometimes be negative, signifying that the circuit is supplying power rather than consuming it. This scenario is feasible in circuits with dependent sources.