Thevenin’s Theorem states that any linear, bilateral electrical network with voltage sources and resistances can be replaced by an equivalent circuit consisting of a single voltage source (V_{th}) in series with a single resistance (R_{th}) connected to a load.

Thevenin’s Theorem is an analytical method used to simplify complex circuits into a simple equivalent circuit. This equivalent circuit consists of a single voltage source in series with a resistance. By transforming the original complex circuit, Thevenin’s Theorem makes it easier to analyze the behavior of the circuit, especially when different loads are connected to it.

## Understanding Thevenin’s Theorem.

Thevenin’s Theorem states that any linear circuit with multiple voltages and resistances can be replaced by an equivalent circuit with a single voltage source (V_{th}) in series with a single resistance (R_{th}). This equivalent circuit is connected across the load.

This theorem is particularly useful for simplifying the analysis of power or battery systems and interconnected resistive circuits. It effectively reduces a complex “one-port” network into a simple two-terminal circuit, making calculations more straightforward.

## Step-by-Step Procedure of Thevenin’s Theorem.

**Identify the Portion of the Circuit:**- Select the portion of the circuit for which you want to find the Thevenin equivalent. Typically, this involves choosing the two terminals across which you will find the Thevenin equivalent.

**Find the Open Circuit Voltage (V**_{OC}or V_{th}):- Remove the load resistor (if there is one) across the chosen terminals.
- Calculate the voltage across these open terminals. This voltage is known as the open-circuit voltage (V
_{OC}), and it will be your Thevenin voltage (V_{th}).

**Find the Thevenin Equivalent Resistance (R**_{th}):- Deactivate all independent sources in the circuit:
- Replace all independent voltage sources with short circuits (i.e., replace each voltage source with a wire).
- Replace all independent current sources with open circuits (i.e., remove the current sources, leaving a break in the circuit).

- Calculate the equivalent resistance seen from the open terminals where the load resistor was connected. This equivalent resistance is the Thevenin resistance (R_th).

- Deactivate all independent sources in the circuit:
**Reconstruct the Thevenin Equivalent Circuit:**- Draw the Thevenin equivalent circuit as a single voltage source (V
_{th}) in series with a single resistance (R_{th}). - Reconnect the load resistor to the Thevenin equivalent circuit.

- Draw the Thevenin equivalent circuit as a single voltage source (V
**Analyze the Simplified Circuit:**- Use the Thevenin equivalent circuit to analyze the current through and the voltage across the load resistor.

## Thevenin’s Theorem Example.

Find the C Voltage as shown in below circuit by using Thevenin’s Theorem.

**Step 1: Identify the Portion of the Circuit:**

Remove the load resistor R_{L} (4Ω).

**Step 2: Find the Open Circuit Voltage (V_{OC} or V_{th}):**

By using KVL for upper loop.

12i_{1} + 8i_{1} + 5 (i_{1} – i_{2}) = 0

25i_{1} – 5i_{2} = 0 Equation-1

By applying KVL around lower loop.

5 (i_{2} – i_{1}) + 20i_{2} = 72

-5i_{1} + 25i_{2} = 72 Equation-2

By solving above 2 equations we got the current values.

i_{1} = 0.6 A , i_{2} = 3 A

so finally ** V_{OC} or V_{th}** will be

**V _{OC} = 8i_{1} + 20 i_{2} = 64.8 V.**

**Step 3: Find the Thevenin Equivalent Resistance ( R_{th}):**

- Deactivate all independent sources.
- Replace Voltage source with a short circuit.
- Calculate the resistance seen from the open terminals.

R_{th} = (12 x 12 ) / (12 + 12) = 144/ 24 = 6Ω

**Step 4: Reconstruct the Thevenin Equivalent Circuit:**

Reconnect the load resistor *R _{L}*=4Ω to the Thevenin equivalent circuit:

As we know that I = V / R

V = ** V_{OC}** and R = R

_{th}+ 4Ω

So, I_{o} = 64.8 / (6 + 4) = 6.48 A

**V _{0} = I_{o} x R = 6.48 x 4 = 25.92 V**

Thevenin’s Theorem simplifies the analysis of complex circuits by reducing them to a simple equivalent circuit. This method involves finding the open-circuit voltage, calculating the equivalent resistance with all independent sources deactivated, and reconstructing the circuit with a single voltage source and resistance. This simplification is particularly useful for analyzing circuits with varying loads, making the analysis more manageable and intuitive.