Thevenin’s Theorem

Thevenin's theorem states that a complex linear network in an electrical circuit will be reduced to a single equivalent circuit. This makes the circuit easier to compute by determining the current in one load resistor and the output voltage across it. Based on this theorem, the practical application of Thevenin's theorem is usually in electrical engineering for simplified circuit analysis with multiple sources and resistors.

1.0Thevenin’s Theorem Statement 

According to Thevenin's theorem, any linear, bilateral network of resistors and sources can be replaced by a single equivalent circuit consisting of:

• A single voltage source referred to as the Thevenin voltage, V^th, placed in series with
• A single resistance (R^th) is known as the Thevenin resistance.
• The load resistance can also be connected to this simplified circuit. 

This is a simplified circuit that makes it easier to analyze the load in the system.

There is one more theorem that is similar to Thevenin theorem named Norton theorem. Let’s understand that too. 

2.0Norton and Thevenin Theorem

Norton and Thevenin's Theorem is essentially similar. Both represent an attempt to reduce a difficult linear circuit into simpler equivalent circuits:

• Thevenin's Theorem reduces a complicated circuit to an equivalent circuit with a voltage source in series with a resistor.

• Norton's Theorem simplifies a complicated circuit to an equivalent circuit with a current source in parallel with a resistor.

The two theorems are interchangeable because you can transform a Thevenin equivalent to a Norton equivalent and vice versa using the following relations:

• Thevenin voltage (V^th) = Norton current (I_N) × Norton resistance (R_N).
• Thevenin resistance (R^th) = Norton resistance (R_N).

Hence, Norton and Thevenin's Theorem provide two different but equivalent methods of representing and reducing the circuit.

3.0Thevenin Theorem Steps

In order to obtain the Thevenin equivalent circuit between two terminals of an existing network, the Thevenin theorem is followed by these steps:

Step 1: Finding Thevenin Voltage (V^th)

• If the circuit contains a load resistance, remove it.
• Now calculate the open-circuit voltage between the two terminals at which the load resistance was connected. This calculated voltage is the Thevenin voltage (Vth).

Step 2: Finding Thevenin Resistance (R^th)

Deactivate all independent sources in the circuit:

• All independent voltage sources are replaced with short circuits (i.e., replace voltage sources with a wire).
• All independent current sources are replaced with open circuits (i.e., take off current sources and act as though no current is passing).
• Calculate the equivalent resistance observed at the two terminals at which the load was attached in the first place. That resistance is the Thevenin resistance (Rth).

Step 3: Replace the Load Resistance with Thevenin Resistance

• With Vth and Rth calculated, attach the load resistance back to the Thevenin equivalent circuit.

4.0Thevenin Theorem Example 

Imagine that a circuit with a voltage of (V) is connected with a series combination of two resistors, let’s say R₁ and R₂, and a load resistor (RL) across two terminals, that is, A and B. 

1. Thevenin Voltage (V^th):

• Disconnect the load resistor (RL).
• Compute the open-circuit voltage across terminals A and B. This is the Thevenin voltage.

2. Thevenin Resistance (R^th): 

• Short circuit the voltage source (replace it with a short circuit).
• Determine the equivalent resistance by looking into terminals A and B. This is the Thevenin resistance.

3. Reconnect the Load Resistor (RL) to the Thevenin equivalent circuit:

• Use the reduced circuit to determine the current or voltage across RL using Ohm's law.

5.0Application of Thevenin theorem 

Thevenin's theorem is widely used in simplifying circuit analysis, especially in the following cases:

• On Simplification of Power Systems: circuits that have numerous components and resistances.
• Maximum Power Transfer: Determine the load resistance that maximizes power delivered to the load, which occurs when load resistance equals Thevenin resistance.
• Reduction of Complex Networks: To transform voltage and current calculations into simpler versions by reducing complex networks into a single voltage source and a single resistor.

6.0Solved Problems

Problem 1: A 10 V voltage source, in series with resistors R₁​=4Ω and R₂=6 Ω, with terminals A and B. Find Thevenin Voltage(V^th).

Solution:

Remove R₂ and calculate the voltage of open-circuit across terminals A and B: 

$V_{th}=10\times \frac{R_2}{R_1+R_2}=6V$

Now, find thevenin resistance (R^th) by deactivating the voltage source. The equivalent resistance will be: 

Rth = R₁ = 4Ω

Problem 2: A 12V voltage source is in series with resistors R₁=3 Ω and R₂ =12 Ω, with load resistor RL=6 Ω across R₂.

Solution: Find Thevenin Voltage (Vt^t) 

Remove R_L and calculate the open circuit voltage. 

$V_{th}=12\times \frac{R_2}{R_1+R_2}=9.6V$

Now, find the resistance by deactivating the voltage. The equivalent resistance is: 

Rth = R₁ ∥ R₂ = 2.4Ω

Problem 3: A 12V battery is connected in series with resistors R₁=4 Ω and R_2​=6Ω. The load resistor R_L=8 Ω is connected in parallel with R₂​. Find the Norton equivalent resistance for the circuit across the load resistor R_L​.

Solution: for Norton Equivalent resistance first remove load resistance R_L. After removing load resistance: 

According to the question R_L is connected to R₂ in parallel combination. 

$\frac{1}{R_{Parallel}}=\frac{1}{R_L}+\frac{1}{R_2}=\frac{1}{8}+\frac{1}{6}$

$\frac{1}{R_{Parallel}}=\frac{6+8}{48}=\frac{14}{48}$

$R_{Parallel}=3.43\Omega$

Now, R₁ and $R_{Parallel}$ is combined in series so: 

$R_N=4+3.43=7.43\Omega$