Thevenin’s Theorem

Definition

Thevenin’s theorem states that any linear, bilateral, two-terminal network of sources and resistances can be replaced by an equivalent circuit consisting of a single ideal voltage source $V_{th}$ in series with an equivalent resistance $R_{th}$, where $V_{th}$ is the open-circuit voltage across the terminals and $R_{th}$ is the resistance seen looking back into the network with all independent sources deactivated.

This equivalent circuit produces the same terminal voltage and current behavior for any load connected to the two terminals.

Main Content

1. Thevenin Equivalent Voltage $V_{th}$

• The Thevenin voltage is the open-circuit voltage across the two terminals of interest.
• It is obtained by removing the load from the circuit and finding the voltage at the open terminals. Since no current flows through the open terminals, this voltage represents the supply potential available to the load.

For example, if a network connected to terminals A and B gives an open-circuit voltage of 12 V, then $V_{th} = 12\text{ V}$. This means the entire network can be represented by a 12 V source in the equivalent circuit.

The Thevenin voltage is important because it tells us the maximum terminal voltage the network can provide when nothing is connected to it. In practical terms, it acts like the “resting voltage” of the circuit before a load is attached.

2. Thevenin Equivalent Resistance $R_{th}$

• The Thevenin resistance is the equivalent resistance seen from the load terminals when all independent sources are turned off.
• Independent voltage sources are replaced by short circuits, and independent current sources are replaced by open circuits. Dependent sources, if any, remain active and must be handled using a test source or circuit analysis methods.

This resistance indicates how much the original network resists current flow when a load is connected. A lower $R_{th}$ generally allows more current to flow through the load, while a higher $R_{th}$ limits current.

For example, if after deactivating the sources the resistance seen from terminals A and B is 4 $\Omega$, then $R_{th} = 4\Omega$. The network is then represented by a 12 V source in series with 4 $\Omega$.

A useful check is that when the load resistance equals $R_{th}$, the load receives maximum power, which is a key result related to Thevenin’s theorem.

3. Load Analysis Using Thevenin’s Theorem

• Once the Thevenin equivalent is formed, the load current, load voltage, and power can be found very easily using Ohm’s law and series circuit rules.
• If the load resistance is $R_L$, then the load current is: $$I_L = \frac{V_{th}}{R_{th} + R_L}$$ and the load voltage is: $$V_L = I_L \cdot R_L$$

This is much simpler than solving the entire original circuit every time a different load is connected. It is especially helpful when the load changes frequently, because the Thevenin equivalent of the source network stays the same.

Example:
If $V_{th} = 12\text{ V}$, $R_{th} = 4\Omega$, and $R_L = 6\Omega$, then: $$I_L = \frac{12}{4+6} = \frac{12}{10} = 1.2\text{ A}$$ $$V_L = 1.2 \times 6 = 7.2\text{ V}$$ So the load current is 1.2 A and the load voltage is 7.2 V.

Working / Process

1. Remove the load from the circuit

Identify the two terminals across which the load is connected and disconnect the load resistor or load element. This leaves the source network intact and allows the open-circuit voltage to be found.

2. Find the open-circuit voltage and equivalent resistance

Calculate the voltage across the open terminals to get $V_{th}$. Then deactivate all independent sources and determine the resistance seen from the same terminals to get $R_{th}$. If dependent sources are present, keep them active and use a test source if needed.

3. Replace the original network with the Thevenin equivalent and reconnect the load

Draw the simplified circuit as a single voltage source $V_{th}$ in series with $R_{th}$, then reconnect the load resistor $R_L$. Use simple series circuit formulas to find load current, load voltage, and power.

Advantages / Applications

• It reduces a complicated linear circuit to a very simple equivalent circuit, making analysis faster and easier.
• It is highly useful for studying the effect of changing loads without reanalyzing the entire network each time.
• It is widely used in power systems, electronic circuits, communication networks, and practical design work for load matching and power transfer.

Summary

• Thevenin’s theorem simplifies any linear two-terminal network into a single voltage source and series resistance.
• The equivalent voltage is the open-circuit terminal voltage, and the equivalent resistance is the resistance seen from the terminals with independent sources deactivated.
• It is mainly used to find load voltage, load current, and power quickly and efficiently.