Thevenin’s Theorem: Working and Calculations

With the help of Thevenin’s Theorem, we can simplify a complicated circuit with several resistors and sources into a much simpler “equivalent circuit.” The total voltage and total resistance of the original circuit are represented by a single voltage source and resistor, respectively, in this analogous circuit.

In the past three tutorials, we have looked at ways to solve difficult electrical circuits. Following our study of Kirchhoff’s Laws, we moved on to mesh analysis and nodal analysis. But the toolbox isn’t finished yet! Numerous “Circuit Analysis Theorems” are available that provide different approaches for determining voltages and currents at every given location in a circuit.

Thevenin’s Theorem, one of the most used methods for circuit analysis outside of Kirchhoff’s Laws, is covered in this lesson. We may simplify complicated circuits and make them easier to analyze thanks to this useful theorem.

Thevenin’s Theorem: Minimizing the Complexity

For circuit analysis, Thevenin’s Theorem provides a potent tool for simplification. It states that any linear circuit can be replaced with a much simpler equivalent circuit, regardless of how complicated it is (with many voltages and resistances). This analogous circuit is made up of:

One voltage source: This source reproduces the circuit’s overall voltage behavior.
A single resistor, also known as an impedance in AC circuits, serves as a symbol for the circuit’s total resistance.

Imagine reducing a disordered system of wires, resistors, and voltage sources to nothing more than a series connection between a voltage source and a resistor! This is Thevenin’s Theorem in its simplest form.

The theorem is especially useful for resistive circuit analysis and power or battery systems analysis. It helps us see how an external load would be impacted by the original complex circuit. We can more effectively study the behavior of the circuit by concentrating on these two essential components: the voltage source and resistance.

Thevenin’s Theorem provides an insightful viewpoint for examining circuits that include external loads. According to this, any complicated “one-port” network (one that has only one pair of input/output terminals) made up of several resistors and energy sources may be replaced by a simpler equivalent circuit that only has two parts. This is because the load resistor (R_L) sees things this way:

• Equivalent Resistance (R_s): This stands for Thevenin resistance, which is the total resistance as seen from the load terminals “looking back” into the circuit (R_L). Stated differently, it records the total resistance that the intricate circuit exhibits to the external load.
• Equivalent Voltage Source (V_s): This is the Thevenin voltage, which is the voltage across the complicated circuit’s terminals when there is no load attached (RL is open). It represents the overall voltage behavior of the circuit in the absence of a load.

When considering the load resistor R_L, the initial complicated network may be reduced to a single resistor (R_s) connected in series with a single voltage source (V_s). This simplification is especially helpful in understanding how the original circuit would effect the associated load when assessing power or battery systems and related resistive circuits.

Take the circuit studied in the earlier lessons, for instance. We can convert that circuit into a much simpler equivalent circuit with simply R_s and V_s by using Thevenin’s Theorem. We may concentrate on these two essential components and examine the behavior of the complex circuit when it is linked to the load resistor RL thanks to this analogous circuit.

The circuit must be set up for an examination of its fundamental resistance (without the load) in order to use Thevenin’s Theorem. Here’s how we go forward:

• Separate the A and B load terminals: The real load resistor that is connected between terminals A and B should be removed first. In this example, this resistor is a 36 Ω resistor.
• Deactivate Internal Sources: We must turn off any internal voltage or current sources in order to isolate the circuit’s internal resistance. This includes:
• Voltage Sources: Every voltage source (shown by the letter “v” in the text) in the circuit has been short-circuited. Envision attaching a wire to the positive and negative terminals of every source to essentially reduce their voltage to zero (v = 0).
• Current Sources: Any source of current (designated by the letter “i”) is regarded as an open circuit. In order to achieve zero current flow (i = 0), a break must be made in the circuit route where the current source was placed.
• Deactivating these sources will produce a “ideal” situation that may be used to determine the equivalent resistance. We may concentrate only on the resistance of the remaining circuit elements by eliminating their impact.
• Calculating Thevenin Resistance (R_s): We may examine the resulting simpler circuit once the terminals (A and B) are separated and the internal sources are turned off. Next, with all voltage sources shorted as previously said, calculate the total resistance “looking back” from terminals A and B to find the Thevenin resistance (R_s).

The simpler circuit that results from completing these steps is shown in the following diagram. With the sources inactive, this circuit enables us to calculate the equivalent resistance (R_s), which is an essential element of Thevenin’s Theorem.

Calculating the Equivalent Resistance (R_s)

9 Ω resistor connected in parallel with the 18 Ω resistor:

• R_T = (R₁ * R₂) / (R₁ + R₂)
• = 18 * 9 / 18 + 10
• = 5.78 Ω

When the load resistor (R_L) is disconnected, the voltage (V_s) equals the potential difference across terminals A and B. To put it simply, (V_s) is the open-circuit voltage that is measured at these terminals.

Calculating the Equivalent Voltage (V_s).

The two voltages must now be reconnected to the circuit, and since V_S = V_AB, the current flowing through the loop is determined as follows:

• I = V / R
• = 18v – 9v / 18Ω + 9Ω
• = 0.33 Amps

Since both resistors share this common current of 0.3333 amperes (330 mA), the voltage drop across either the 18 Ω or 9 Ω resistor may be calculated as follows:

• V_AB = 18 – (18Ω * 0.3333amps)
• = 12 Volts
• or V_AB = 9 + (9Ω * 0.3333amps)
• = 12 Volts

Following this, a voltage source of 12 volts and a series resistance of 5.78 Ω would make up the Thevenin’s Equivalent circuit. Re-connecting the 36 Ω resistor into the circuit yields the following results:

and based on this, the circuit’s current is calculated as follows:

• I = V / R
• = 12v / 5.78Ω + 36Ω
• = 0.286 amps

This result, as we discovered in the previous circuit analysis lesson, is exactly the same value of 0.286 amps implementing Kirchhoff’s circuit equation.

Thevenin’s Theorem provides yet another effective circuit analysis technique. It works especially effectively when dealing with complex circuits composed of:

• resistors
• One or more sources of voltage
• current resources

There are many combinations of parallel and series connections that may be made with these components.

Although current and voltage correlations may be used to mathematically represent Thevenin’s Theorem, this method might not be the most effective one for very large networks. This is due to the fact that Thevenin’s Theorem frequently uses methods such as Nodal Voltage Analysis or Mesh Current Analysis to get the equivalent resistance (R_s) and voltage (V_s). Under such circumstances, applying Mesh or Nodal Analysis right away might be simpler.

But in circuit design, Thevenin’s Theorem really shines. With its help, we may design similar circuits that are more straightforward for transistors, voltage sources (such as batteries), and other parts. These equivalent circuits seem to be extremely useful resources for circuit design and analysis.

Conclusions:

Thevenin’s Theorem gives us a useful tool for breaking down complex electrical networks. It lets us swap out complicated circuits with much simpler equivalents that just require a few number of resistors and voltage/current sources. This analogous circuit is made up of:

One voltage source (V_s): This source records the original circuit’s overall voltage behavior.
A single resistor (R_s): As seen from the exterior connection points, this resistor reflects the whole resistance of the original circuit.

Consider an interconnected network of sources of electricity, resistors, and cables. We can reduce this complexity to just a voltage source and a resistor linked in series according to Thevenin’s Theorem!

This is where the magic lies: This similar circuit acts exactly like the original complicated circuit it replaces when seen from the corresponding terminals (A and B). Stated otherwise, the relationship between current and voltage (i-v) at terminals A and B stays precisely the same.

Procedure for Using Thevenin’s Theorem:

• Isolating the Load: Take off the component you want to analyze or the load resistor (R_L).
• Identifying Resistance to Thevenin (Rs): Replace every source of voltage with a short circuit and every source of current with an open circuit. Next, figure out how much resistance there is overall “looking back” from terminals A and B.
• Calculating the Voltage (V_s) of Thevenin: Reactivate the circuit’s sources of current and voltage (without the load). Now, use common circuit analysis techniques to determine the voltage between terminals A and B.
• Analyzing the Load: Using this simplified equivalent circuit, you can use R_s and V_s to examine the current passing through the original load resistor (R_L).

Another effective simplification method, Norton’s Theorem, will be covered in the next lesson. It gives us a new angle on circuit analysis by enabling us to depict a complicated circuit with a single current source connected in parallel to a single resistance.

References: Thévenin’s theorem

Thevenin’s Theorem