What Is Thevenin’s Theorem?: DC Circuit Analysis

Thevenin’s theorem is an analytical method that simplifies complicated circuits containing multiple resistors and sources into a single voltage source in series with a single resistor. When applied to DC circuits, it helps clarify internal resistances and the currents flowing through load resistors. For instance, if you feel it’s challenging to grasp all the voltages and currents in a circuit that has several power supplies and resistors, Thevenin’s theorem can reduce the number of calculation steps. Even in everyday situations like switch circuits or power supply circuits, understanding this principle can significantly improve circuit analysis, which is why it’s widely employed in both learning and professional contexts. From here on, we will organize and explain the fundamental concepts of Thevenin’s theorem, how to introduce it, and the steps to carry out actual calculations.

Overview of Thevenin’s Theorem

By understanding an overview of Thevenin’s theorem, you can gain a clearer picture of DC circuit analysis as a whole. Its property of reducing complicated circuits into an equivalent, simple form has been adopted by many electrical and electronics engineers. For example, even if you have a network with numerous resistors, converting it to a Thevenin voltage and a Thevenin resistor can make it easier to determine the load current. Below is a concrete explanation of the position of this theorem in circuit theory.

Key Elements of Thevenin’s Theorem

When you study Thevenin’s theorem, it’s essential to understand what Thevenin voltage and Thevenin resistance mean. These two parameters capture how a complicated circuit behaves in the form of “one voltage source + one resistor.” You can easily compute the current and voltage once a load is connected by appropriately determining the open-circuit voltage at the terminals and combining the resistors after removing independent sources. The following sections focus on calculating the Thevenin voltage and Thevenin resistance, and how to handle load resistors.

Open-Circuit Voltage: What Is the Thevenin Voltage (V_Th)?

The Thevenin voltage (V_Th) refers to the voltage value of the single equivalent source in a circuit simplified by Thevenin’s theorem. Practically, it is found by opening the two terminals in question and measuring or analyzing the potential difference that appears there. Specifically, remove the load from the circuit and leave the two relevant terminals open. The voltage measured or computed across those open terminals is the Thevenin voltage. A standard procedure for finding it is:

1. Remove the load resistor or external circuit
   • Keep the rest of the circuit as is, but remove the load at the two terminals.
2. Calculate or measure the potential at the open terminals with the sources and resistors still connected
   • Apply Ohm’s law, Kirchhoff’s voltage law, and Kirchhoff’s current law to find the actual voltage across the open terminals.

This way, you discover that the terminals carry some voltage even without a load resistor and that voltage is taken to be V_Th, which becomes the “single voltage source” in the Thevenin equivalent circuit.

Even if the circuit has two or more voltage sources in series or parallel, you can measure a single consistent voltage when you only look at the output terminals. If the circuit is linear, you can apply the principle of superposition or straightforward voltage division to get that voltage. Moreover, finding the open-circuit voltage is straightforward, even if the circuit looks complicated.

How to Determine the Thevenin Resistance (R_Th)

Alongside the Thevenin voltage, the Thevenin resistance (R_Th) is a critical element. In the Thevenin equivalent circuit, R_Th is placed in series with V_Th, describing how the circuit’s two terminals appear regarding resistance. The standard method to find R_Th is as follows:

1. Remove independent sources from the circuit
   • For an ideal voltage source with zero internal resistance, replace it with a short circuit. For an ideal current source, replace it with an open circuit.
   • If there are dependent sources (for example, voltage-controlled current sources or voltage-controlled voltage sources), keep them in place and use other techniques (such as a test source method) to determine the resistance.
2. Compute the apparent resistance across the two terminals
   • Combine resistors in series and parallel to find the single equivalent resistance.
   • If dependent sources exist, you might use the test source method, injecting a test voltage or current and analyzing the result to find R_Th.

In essence, R_Th summarizes how the network of resistors is arranged. For example, short-circuiting an ideal voltage source can turn the circuit into a simpler combination of parallel and series resistors. You can use the standard equations for series and parallel resistor combinations if no dependent sources are present.

Calculating Current and Voltage in Thevenin’s Equivalent Circuit (When a Load Is Connected)

After obtaining the Thevenin equivalent circuit, you only need to consider V_Th, R_Th, and the load resistor RL to analyze the overall circuit. Typically, you calculate:

\(I_L=\displaystyle \frac{V_{Th}}{R_{Th}+R_L}\)

\(V_L=I_L×R_L\)

I_L is the current flowing through the load resistor, and V_L is the voltage drop across it. Even if the original network appeared complicated, once you find the two parameters V_Th and R_Th, you can rely on these straightforward formulas to determine current and voltage. This reduction eases the workload for circuit designers and makes it easier to conceptualize the circuit as “one large power source and one resistor in series.”

Procedure for Applying Thevenin’s Theorem

When practically applying Thevenin’s theorem to a circuit, it’s crucial to follow the fundamental steps precisely. Identifying which two terminals to focus on, how to find the open-circuit voltage, and how to combine resistors once the sources are removed—understanding these parts of the process will greatly simplify DC circuit analysis that might otherwise seem daunting. Below, we detail the process of employing Thevenin’s theorem through specific steps and examples.

Assumptions and Important Points in Linear Circuits

Thevenin’s theorem assumes that the circuit in question is linear. A linear circuit includes linear components like resistors, inductors, and capacitors, along with independent and dependent sources. Circuits that contain nonlinear elements such as diodes or transistors can sometimes be approximated by linearization around an operating point, but additional considerations may be necessary.

In DC circuits, inductors and capacitors may be treated as short or open circuits under certain conditions, making Thevenin’s theorem easier to apply. For instance, under steady-state DC conditions, a capacitor behaves like an open circuit, and an inductor behaves like a wire. Therefore, you may only need to focus on resistors and voltage or current sources, making the procedure more straightforward. However, considering transients or AC components, you must understand Thevenin’s theorem’s applicable scope.

The Step-by-Step Process

Here is a concise overview of how to calculate with Thevenin’s theorem:

1. Identify the two terminals of interest
   • Be clear about which load or measurement terminals you’re focusing on.
2. Remove the load resistor and find the open-circuit voltage (VOC)
   • This open-circuit voltage is your Thevenin voltage, V_Th.
3. Remove independent sources from the circuit and combine resistors to find R_Th
   • Voltage sources become short, and current sources become open. Use methods like the test source approach to calculate R_Th if dependent sources exist.
4. Draw a single circuit with V_Th in series with R_Th (the Thevenin equivalent)
5. Reconnect the load resistor (R_L) as a simple circuit with one voltage source, one series resistor, and the load

The critical factors determining the open-circuit voltage and handling the independent sources when computing the equivalent resistance. Doing these correctly ensures that Thevenin’s theorem yields accurate results. Moreover, you typically don’t have to worry about complex frequency-dependent behavior in DC circuits, so finding V_Th and R_Th usually proceeds smoothly.

A Concrete Example with Multiple Resistive Circuit Elements

Let us illustrate Thevenin’s theorem by simplifying a sample DC circuit. For example, consider the following:

• Power supply: A DC voltage source E = 12V
• Resistors: Three resistors R₁ = 6Ω, R₂ = 3Ω, R₃ = 2Ω
• Connections:
  • The positive terminal of the supply goes through R₁.
    R₂ and R₃ are connected in parallel from the node after R₁ (call it Node A) down to the negative terminal (ground).
    A load resistor R_L can also be attached from Node A to the ground (i.e., the negative terminal), and we can remove R_L to observe the circuit at those two terminals.

Though it may sound complicated in words, it essentially means:

• The node after R₁ is Node A.
• From Node A to ground, R₂ and R₃ are always connected in parallel.
• The load resistor R_L can be connected or removed between Node A and the ground.

1. Finding the Open-Circuit Voltage

To apply Thevenin’s theorem, we remove the load resistor R_L. We want the voltage between Node A and the ground (the negative terminal). With R_L removed, the circuit still has current flowing from the 12V source, through R₁, and then through the parallel combination of R₂ and R₃ to the ground. Hence, Node A’s voltage is determined by voltage division between R₁ and (R₂‖R₃).

・Calculate the parallel of R₂ and R₃:

\(R_2\|R_3=\displaystyle \frac{3×2}{3+2}=1.2Ω\)

・Total series resistance from the source to the ground:

\(R_{total}=R_1+(R_2\|R_3 )=6+1.2=7.2Ω\)

・Current through R₁ and the parallel branch:

\(I_L=\displaystyle \frac{E}{R_{total}}=\displaystyle \frac{12}{7.2}≈1.6667A\)

・Voltage at Node A (the open-circuit voltage):

\(V_{Th}=I_L×(R_2\|R_3)=2.0V\)

Therefore, the open-circuit voltage at Node A with respect to ground (where R_L was removed) is:

2. Finding the Thevenin Equivalent Resistance (R_Th)

Next, to find R_Th, replace the ideal voltage source with a short circuit (i.e., a wire) and look back into the circuit from Node A to the ground:

• With the supply shorted, Node A “sees” R₁, R₂, and R₃ all connected to the ground—hence, they are in parallel from Node A’s perspective.

Thus,

\(R_{Th}=R_1\|R_2\|R_3=6\|3\|2\)

First compute 6 ‖ 3:

\(6\|3=\displaystyle \frac{6×3}{6+3}=\displaystyle \frac{18}{9}=2Ω\)

Then compute 2 ‖ 2 (the result of (6 ‖ 3) in parallel with R₃=2Ω):

\(2\|2=\displaystyle \frac{2×2}{2+2}=1Ω\)

Hence,

\(R_{Th}=1Ω\)

3. Final Thevenin Equivalent Circuit

Summarizing our results:

V_Th = 2V, R_Th = 1Ω

Once the load resistor R_L is reconnected between Node A and ground, the circuit can be viewed as a series connection of R_Th and R_L across a 2.0V source:

\(I_L=\displaystyle \frac{V_{Th}}{R_{Th}+R_L}=\displaystyle \frac{2}{1+R_L}\)

Thus, using Thevenin’s theorem, what was originally a multi-resistor circuit becomes a simple two-element series circuit, dramatically simplifying the load current calculation.

Thevenin’s Theorem and Maximum Power Transfer

After deriving an equivalent Thevenin circuit, you may want to know how much power can be delivered to the load. By applying the Maximum Power Transfer Theorem, you can find an optimal operating point by looking at the relationship between the load and Thevenin resistance. This section clarifies the basics of maximum power transfer and explains how it’s practically used in circuit design.

Overview of Maximum Power Transfer

When studying Thevenin’s theorem, you often encounter the “Maximum Power Transfer” theorem, which states that maximum power is delivered to the load if the load resistance R_L equals the Thevenin resistance R_Th. Interpreting the circuit as a simple Thevenin equivalent makes it intuitive to see why that’s the case.

\(P_L=I_L^2 R_L=\left(\displaystyle \frac{V_{Th}}{R_{Th}+R_L}\right)^2 R_L\)

To maximize this power with respect to R_L, you set

\(R_L=R_{Th}\)

This is commonly used in applications such as communications equipment, antenna matching, and audio amplifiers, where you might adjust the load characteristics.

Practical Applications and Caveats

While maximum power transfer is helpful if your goal is to maximize power consumed by the load, it can also lead to increased current and heat dissipation in the circuit. In real settings, you might not be free to alter the supply voltage or other circuit conditions, so R_L = R_Th might not be feasible. You must balance efficiency, thermal design, and component ratings in practical design. Still, it helps to know that the maximum power transfer concept is directly tied to Thevenin’s theorem.

Relation Between Thevenin’s Theorem and Other Circuit Theorems

Looking at circuit theory broadly, Thevenin’s theorem doesn’t exist in isolation. Norton’s theorem is its counterpart, and along with the principle of superposition and Kirchhoff’s laws, Thevenin’s theorem addresses a wide range of problems. Here, we will explore how Thevenin’s theorem connects with other fundamental circuit theorems.

Norton’s Theorem

Norton’s theorem goes hand in hand with Thevenin’s. It states that “a single current source in parallel with a single resistor can equivalently represent any linear circuit.” While Thevenin’s theorem uses “voltage source + series resistor,” Norton’s theorem uses “current source + parallel resistor.”

They are interconvertible: from V_Th and R_Th, you can get

\(I_N=\displaystyle \frac{V_{Th}}{R_{Th}}\)

Thus, the source can be seen as a current source I_N = V_Th / R_Th in parallel with R_Th. Which form you prefer often depends on whether your circuit arrangement is branching or simpler in series.

Conversion to Norton Equivalent (Newly Added Section)

When you already have a Thevenin equivalent circuit, converting it to its Norton equivalent is straightforward. Suppose your Thevenin equivalent is described by:

V_Th in series with R_Th

To convert this into the Norton form (a current source I_N in parallel with a resistor R_N):

1. Calculate the Norton current I_N

\(I_N=\displaystyle \frac{V_{Th}}{R_{Th}}\)

This ensures the same open-circuit voltage and short-circuit current as the Thevenin circuit.

2. Use the same resistor value

\(R_N=R_{Th}\)

The parallel resistor in the Norton circuit is numerically equal to the series resistor in the Thevenin circuit.

3. Draw the new circuit
   • Replace the Thevenin voltage source V_Th with a current source of magnitude I_N.
   • Put R_N parallel with that source instead of in a series.

This conversion allows you to analyze specific branching configurations more easily. For instance, if you’re dealing with multiple parallel loads, a Norton model (current source + parallel resistor) can sometimes simplify the calculations of parallel current splits.

Superposition and Kirchhoff’s Laws

The derivation and application of Thevenin’s theorem are underpinned by the superposition principle and Kirchhoff’s voltage and current laws. Thevenin’s theorem states that multiple sources in a linear circuit can be combined into one source due to linearity and superposition.

• Superposition Principle: You can turn on each source individually and add up the resulting voltages and currents (for linear circuits).
• Kirchhoff’s Voltage Law (KVL) / Kirchhoff’s Current Law (KCL): Fundamental circuit rules can be used in node or loop analysis.

When applying Thevenin’s theorem, you might use superposition to find the open-circuit voltage and KCL or KVL to combine resistors after removing the sources. Recognizing these steps clarifies that Thevenin’s theorem relies on standard circuit fundamentals.

Practical Examples of Thevenin’s Theorem in DC Circuits

Seeing how Thevenin’s theorem applies in real-world design and analysis helps deepen your understanding. Whether it’s power supply circuits, sensor and actuator circuits, or motor driver circuits, simplifying the complicated parts into an equivalent circuit has significant advantages. Below are some specific scenarios focusing on DC circuits where Thevenin’s theorem can be effectively used.

Load Variation Analysis in Power Supply Circuits

Consider a power supply unit where you connect various resistors or devices as the load, checking how the voltage and current shift. Even if the power supply’s internal circuit is complex, from the viewpoint of its two output terminals, you can approximate it as a Thevenin equivalent. Then, as you change the load, the load voltage and current can be derived simply from V_Th + R_Th + R_L. This speeds up design evaluations for the power supply.

Simplifying Sensor and Actuator Circuits

Sensors and actuators often have clearly defined input or output terminals that connect to other devices. Even if the sensor’s internal resistance, biasing, or power circuits are complex, you can represent the sensor output side as a single voltage source plus a series resistor. This simplifies calculating how the sensor output voltage will appear at an amplifier or microcontroller input.

Such simplifications help predict signal levels or operating points, though if the circuit is highly nonlinear or has significant offsets, the designer must be mindful of additional details. Still, the fundamental advantage is clear: treat the internal details as “one 2-terminal world,” letting you conceptualize it as a straightforward model.

DC Motor and Heater Drive Circuits

Although DC motors and heaters may seem straightforward, their load characteristics can change with temperature or speed. Sometimes, you can approximate the behavior at the motor terminals by a kind of internal voltage source plus series resistance. Using a Thevenin-like model for a given speed range, you can estimate the current drawn under different load conditions. While a motor does have nonlinear characteristics, if you examine a specific operating range, Thevenin’s approach can be used to find a workable approximation.

Points to Keep in Mind for Reliable Calculations

Although Thevenin’s theorem greatly simplifies circuit representation, real circuits may contain dependent sources, nonlinear components, and all sorts of parasitic elements. Misjudging how much idealization or what compensations to include can lead to significant discrepancies between theory and practice. This section summarizes the primary considerations when relying on Thevenin’s theorem.

Handling Dependent Sources

Some circuits include dependent sources, such as voltage-controlled current sources and voltage-controlled voltage sources. In such cases, shorting or opening the independent voltage or current sources may not be enough to determine R_Th. You can’t remove them outright because dependent sources vary according to other voltages or currents.

That’s where the test source method comes in. You place a notional voltage or current source across the two terminals in question and analyze the resulting current or voltage, using their ratio to find the equivalent resistance:

• Apply a test voltage V_test and measure the current I_test.
• Then R_Th = V_test / I_test.

This approach works because linearity still holds if the circuit is fundamentally linear, even with dependent sources.

Transient or High-Frequency Behavior

Thevenin’s theorem can also be adapted to AC or transient analysis, but inductors and capacitors become frequency-dependent impedances rather than simple resistors. Also, parasitic capacitances and inductances can’t be ignored at high frequencies, meaning you might need a more advanced approach or complex notation for impedances.

These frequency considerations usually don’t apply in purely DC analysis, making Thevenin’s theorem straightforward. However, if dealing with motors, heaters, or other loads whose conditions change over time or temperature, you need to think carefully about how valid the linear approximation is for your specific goal.

Reference Article: Electrical impedance

Limitations of Thevenin’s Theorem

While Thevenin’s theorem significantly streamlines circuit analysis, there are several limitations and conditions under which its direct application may not be sufficient:

1. Nonlinear Components and Regions
   Suppose a circuit contains diodes, transistors, or any strongly nonlinear element operating outside a small-signal region. In that case, Thevenin’s theorem in its basic form will not accurately represent the behavior unless additional approximations or linearization techniques are used.
2. Frequency-Dependent ElementsHigh-frequency or transient cases often require considering capacitive or inductive reactances and parasitic effects. A single resistor is usually insufficient to model the frequency response. Therefore, Thevenin’s theorem must be adapted to an AC or s-domain approach (with impedances) if accurate frequency-dependent behavior is needed.
3. Time-Varying or Switching CircuitsCircuits that operate in a switching mode (for instance, PWM power stages) or that change their topology over time may not be easily represented by a single static voltage source and series resistor. Analysis might demand piecewise linear methods or more complex transient simulations.
4. Dependent Sources RevisitedAlthough they can be handled with test-source methods, dependent sources can complicate the determination of R_Th. In some cases, the circuit’s controlling signals might not be strictly linear relationships, making the fundamental assumptions behind Thevenin’s theorem less straightforward to apply.
5. Practical Measurement and TolerancesIn real-world scenarios, measuring V_Th and R_Th might not match theoretical values due to component tolerances, wiring resistances, and other losses. Even a slight discrepancy can matter in precision circuits or high-power designs.

Despite these limitations, Thevenin’s theorem remains a robust conceptual and calculation tool. Recognizing when and how to account for nonlinearities, time variations, or high-frequency effects ensures that the simplified model offers valuable insights.

Reconciling Measurements with Theoretical Values

Paper-based calculations or simulations yield a certain Thevenin equivalent, but actual measurements may differ. Tolerances in components, wiring, contact resistances, and other real-world factors can create discrepancies. You might conclude, “The Thevenin voltage of this circuit is about 11.8V, and the Thevenin resistance is around 3.1Ω,” for example, based on real tests. Adjusting your design or analysis to reflect actual conditions often leads to more reliable outcomes.

Conclusion

Thevenin’s theorem is a handy tool for simplifying DC circuit analysis. It states that a circuit containing multiple resistors and sources can be represented, from the perspective of two terminals, as a single voltage source V_Th in series with a single resistor R_Th. This makes it easier to recalculate currents and voltages when changing the load or carrying out other manipulations.

Additionally, Thevenin’s equivalent circuit forms the basis for dealing with optimization problems such as maximum power transfer. One crucial viewpoint in actual circuit design or troubleshooting is isolating a seemingly complicated network into a workable portion. Thevenin’s theorem reduces everything to a simpler model with just the proper terminals identified.

Because Thevenin’s theorem intertwines with other fundamental theorems—like Norton’s theorem, superposition, and Kirchhoff’s laws—it’s a crucial step in developing a broader understanding of electrical circuits. It’s especially convenient in DC analysis if the circuit mainly consists of resistors. Extra care or additional modeling might be required for high-frequency or nonlinear elements. Still, the core idea remains: “Focus on the two terminals’ behavior, and you can treat the circuit as one source and one resistor.”

If you remember Thevenin’s theorem, you’ll be able to group numerous circuit elements into a single Thevenin equivalent in your mental model, simplifying load design and power supply adjustments later on. Even when learning, practicing calculations on paper with a few resistors and real measurements of open-circuit voltages and short-circuit currents helps you appreciate its usefulness.

By understanding and correctly applying Thevenin’s theorem, you can simplify DC circuit analysis and reduce the risk of mistakes. If you ever look at a circuit diagram and feel unsure where to begin, recalling Thevenin’s theorem and focusing on the two terminals you care about is one valid approach. Ultimately, it clarifies not only the load’s current and voltage but also the circuit’s overall operation.