Thevenin's Theorem Examples

Why This Matters

Thevenin's Theorem lets you reduce any linear circuit, no matter how complex, down to just two components: a single voltage source ($V_{Th}$) in series with a single resistance ($R_{Th}$). This makes analyzing load behavior far simpler and shows up constantly in circuit design and on exams.

The catch is that the method for finding $V_{Th}$ and $R_{Th}$ changes depending on what's in your circuit. Independent sources, dependent sources, and reactive elements each require a different approach. Knowing which technique to use and why is what separates real understanding from 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: replace voltage sources with short circuits and current sources with open circuits. Then calculate the equivalent resistance seen from the load terminals.

Simple Resistive Circuit with a Single Voltage Source

1. Remove the load from the circuit. Leave the terminals open.
2. Find $V_{Th}$ by calculating the open-circuit voltage across those terminals. Use voltage divider or KVL as needed.
3. Find $R_{Th}$ by replacing the voltage source with a short circuit (a wire), then combining series and parallel resistances as seen from the load terminals.
4. Verify: reconnect the load in your Thevenin equivalent and check that $I_L = \frac{V_{Th}}{R_{Th} + R_L}$ matches your original circuit analysis.

Circuit with Multiple Voltage Sources

• Apply superposition: analyze each voltage source one at a time. While you analyze one source, short all the others.
• Sum the contributions from each source to get 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, only on the resistor network.

Circuit with Current Sources

• Use nodal analysis to find the open-circuit voltage across the load terminals. Current sources set up known node equations, making this approach natural.
• Open the current source when finding $R_{Th}$. An open circuit means zero current, which effectively removes the source from the network.
• Convert strategically: sometimes a source transformation (current source in parallel with a resistor → voltage source in series with that resistor) simplifies the circuit before you apply Thevenin's theorem.

Circuit with Both Voltage and Current Sources

• Superposition still works: short voltage sources and open current sources when analyzing each one 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 (find $V_{oc}$ and $I_{sc}$, then $R_{Th} = \frac{V_{oc}}{I_{sc}}$) may be faster than superposition.

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Circuits with Dependent Sources

Dependent sources change the process significantly. You cannot simply turn them off because their value depends on a circuit variable that must remain active. If you deactivate a dependent source, you destroy the relationship it models, and your answer will be wrong.

Finding $V_{Th}$ and $R_{Th}$ with Dependent Sources

For $V_{Th}$:

1. Remove the load and leave the terminals open.
2. Keep all sources (independent and dependent) active.
3. Use KVL, KCL, or mesh/nodal analysis to find the open-circuit voltage. You'll need to express the dependent source's value in terms of a circuit variable and solve the resulting equations simultaneously.

For $R_{Th}$ (test-source method):

1. Deactivate all independent sources (short voltage sources, open current sources). Leave dependent sources in the circuit.
2. Apply a test voltage $V_{test}$ across the open terminals (or inject a test current $I_{test}$).
3. Solve for the resulting $I_{test}$ (or $V_{test}$) using circuit analysis.
4. Calculate $R_{Th} = \frac{V_{test}}{I_{test}}$.

This ratio captures the effect of the dependent source on the equivalent resistance. Sometimes $R_{Th}$ can even come out negative, which is physically meaningful in circuits with active elements.

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Special Circuit Configurations

Some circuits have structures that require a modified approach. 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 the 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. This means $V_{Th} = 0$ across that element, which simplifies analysis dramatically.
• For unbalanced bridges, remove the bridge element, find $V_{Th}$ across its terminals using KVL or nodal analysis, and find $R_{Th}$ by deactivating sources and reducing the resulting resistor network (often two pairs of series resistors in parallel).

Op-Amp Circuit

• Use ideal op-amp assumptions: infinite input impedance (no current into the input terminals), zero output impedance, and the virtual short ($V_+ = V_-$) when negative feedback is present.
• Thevenin equivalent at the output: for an ideal op-amp with feedback, $R_{Th} \approx 0$ at the output, and $V_{Th}$ equals the closed-loop output voltage determined by the feedback network.
• Feedback networks set the closed-loop gain. Analyzing the Thevenin equivalent seen by the feedback path helps you understand gain and loading effects.

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AC and Frequency-Domain Circuits

When capacitors, inductors, or transformers appear, you work in the frequency domain. Thevenin's theorem still applies, but now you find a Thevenin impedance $Z_{Th}$ (a complex number) instead of a simple resistance.

Circuit with Capacitors and Inductors (AC Analysis)

1. Convert to phasors: represent sinusoidal voltages and currents as complex numbers with magnitude and phase angle.
2. Replace components with impedances: $Z_R = R$, $Z_C = \frac{1}{j\omega C}$ for capacitors, and $Z_L = j\omega L$ for inductors.
3. Find $V_{Th}$ (now a phasor) and $Z_{Th}$ (now complex) using the same Thevenin process as DC circuits, but with impedances replacing resistances.
4. $Z_{Th}$ has real and imaginary parts: the real part is resistive, and the imaginary part is reactive. Both affect the 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 side.
• Find the Thevenin equivalent on one side by reflecting all components to that side first, then applying standard methods to the resulting single-loop or single-node circuit.

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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 the input-output relationships of a network without needing to know its internal structure.
• Thevenin equivalent at each port can be derived from these parameters. For example, with the output port open-circuited, the input port Thevenin equivalent has $V_{Th} = Z_{11}I_1$ and $Z_{Th} = Z_{11}$.
• Cascade complex networks by multiplying ABCD matrices of individual stages, which simplifies multi-stage amplifier or filter analysis considerably.

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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?