Thevenin's Theorem Examples

Why This Matters

Thevenin's Theorem is one of the most powerful tools in your circuit analysis toolkit—and it shows up constantly on exams. You're being tested on your ability to reduce complex circuits to simple equivalents, which means understanding when to apply the theorem and how the process changes depending on what's in your circuit: independent sources, dependent sources, reactive elements, or combinations of all three. Mastering these variations is essential for tackling both multiple-choice problems and free-response questions efficiently.

The key insight is that any linear circuit can be reduced to a single voltage source ($V_{Th}$) in series with a single resistance ($R_{Th}$). But the method for finding these values shifts depending on your circuit's components. Don't just memorize the steps for one scenario—know which technique applies to each source type, and understand why dependent sources require a different approach than independent ones. That conceptual understanding is what separates strong exam performance from mere formula plugging.

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Circuits with Independent Sources Only

These are your foundational cases. When a circuit contains only independent voltage and/or current sources, you can find $R_{Th}$ by "turning off" all sources—replacing voltage sources with short circuits and current sources with open circuits—then calculating the equivalent resistance seen from the load terminals.

Simple Resistive Circuit with a Single Voltage Source

• Remove the load and find $V_{Th}$—this is the open-circuit voltage across the terminals where the load was connected
• Short the voltage source to find $R_{Th}$ by combining series and parallel resistances from the load's perspective
• Verify with Ohm's Law: once you have the equivalent, $I_L = \frac{V_{Th}}{R_{Th} + R_L}$ should match your original circuit analysis

Circuit with Multiple Voltage Sources

• Apply superposition—analyze each voltage source independently by shorting all other voltage sources
• Sum the contributions from each source to find the total open-circuit voltage $V_{Th}$
• Calculate $R_{Th}$ once by shorting all voltage sources simultaneously; the equivalent resistance doesn't depend on source values

Circuit with Current Sources

• Use nodal analysis to find the open-circuit voltage across the load terminals
• Open the current source when finding $R_{Th}$—an open circuit means zero current, effectively removing the source
• Convert strategically: sometimes a source transformation (current source to voltage source) simplifies the problem before applying Thevenin's theorem

Circuit with Both Voltage and Current Sources

• Superposition still works—short voltage sources and open current sources when analyzing each independently
• Combine contributions to find $V_{Th}$, then deactivate all sources together to find $R_{Th}$
• Choose your method wisely: for circuits with many sources, direct open-circuit/short-circuit analysis may be faster than superposition

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Circuits with Dependent Sources

Dependent sources change everything. You cannot simply turn them off because their value depends on a circuit variable that must remain active. Instead, apply a test source and calculate the resulting response.

Circuit with Dependent Sources

• Find $V_{Th}$ normally—remove the load and calculate open-circuit voltage using nodal or mesh analysis with dependent sources active
• For $R_{Th}$, apply a test source—deactivate independent sources, then apply a test voltage $V_{test}$ (or current $I_{test}$) at the terminals
• Calculate $R_{Th} = \frac{V_{test}}{I_{test}}$—this ratio gives you the equivalent resistance including the effect of dependent sources

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Special Circuit Configurations

Some circuits have unique structures that require modified approaches. Recognizing these configurations saves time and prevents errors.

Bridge Circuit

• Identify the Wheatstone bridge structure—four resistors arranged in a diamond with a source across one diagonal and load across the other
• Check for balance: if $\frac{R_1}{R_2} = \frac{R_3}{R_4}$, no current flows through the bridge element, simplifying analysis dramatically
• Apply Thevenin's theorem by removing the bridge element and finding $V_{Th}$ across its terminals; unbalanced bridges require full circuit analysis

Op-Amp Circuit

• Use ideal op-amp assumptions—infinite input impedance, zero output impedance, and the virtual short ($V_+ = V_-$)
• Thevenin equivalent at the output is simply $V_{Th} = A_{OL}(V_+ - V_-)$ with $R_{Th} \approx 0$ for ideal op-amps
• Feedback networks determine closed-loop behavior; analyze the Thevenin equivalent seen by the feedback path to understand gain and stability

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AC and Frequency-Domain Circuits

When capacitors, inductors, or transformers appear, you're working in the frequency domain. Thevenin's theorem still applies, but now you're finding a Thevenin impedance $Z_{Th}$ instead of a resistance.

Circuit with Capacitors and Inductors (AC Analysis)

• Convert to phasors—represent voltages and currents as complex numbers with magnitude and phase
• Use impedances: $Z_C = \frac{1}{j\omega C}$ for capacitors and $Z_L = j\omega L$ for inductors; these replace resistances in your calculations
• $Z_{Th}$ is complex—it has both real (resistive) and imaginary (reactive) components, affecting both magnitude and phase of the load response

Circuit with Transformers

• Apply the turns ratio $n = \frac{N_2}{N_1}$: voltage scales by $n$, current scales by $\frac{1}{n}$
• Use reflected impedance—an impedance $Z_L$ on the secondary appears as $\frac{Z_L}{n^2}$ when viewed from the primary
• Find Thevenin equivalent on one side by reflecting all components to that side first, then applying standard methods

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Network Abstraction

For complex systems, two-port network theory provides a systematic framework that builds on Thevenin concepts.

Two-Port Network

• Define port parameters—Z-parameters, Y-parameters, H-parameters, or ABCD-parameters describe input-output relationships
• Thevenin equivalent at each port can be derived from the parameters: $V_{Th} = Z_{11}I_1$ and $Z_{Th} = Z_{11}$ for the input port (with output open)
• Cascade complex networks by converting to two-port representations, simplifying multi-stage amplifier or filter analysis

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Quick Reference Table

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Self-Check Questions

1. What is the key difference in finding $R_{Th}$ for a circuit with only independent sources versus one containing dependent sources?

2. When applying superposition to a circuit with two voltage sources and one current source, how do you deactivate each source type, and why?

3. Compare the Thevenin equivalent process for a DC resistive circuit versus an AC circuit with capacitors—what quantity replaces $R_{Th}$, and how does this affect your calculations?

4. A bridge circuit is balanced when what condition is met? How does this simplify finding the Thevenin equivalent across the bridge element?

5. FRQ-style: Given a circuit with one independent voltage source and one voltage-controlled current source (VCCS), outline the complete procedure for finding both $V_{Th}$ and $R_{Th}$ at a specified load terminal. Why can't you simply short the voltage source and calculate resistance directly?