Thevenin's Theorem: Equivalent Voltage and Resistance

Any linear resistive network, no matter how many sources and resistors it contains, looks identical from the outside to a single voltage source in series with a single resistor — as long as you only care about what happens at one pair of terminals. This is the practical value of Thevenin's theorem: it collapses arbitrarily complex linear circuits into a two-element model, which is exactly what you need when analyzing how a load will behave, designing for maximum power transfer, or estimating output impedance without re-deriving the whole network every time the load changes.

The Theorem

Any linear, bilateral circuit containing independent and/or dependent sources and resistors, viewed from two terminals A and B, can be replaced by an equivalent circuit consisting of:

• A single voltage source V_th (the Thevenin equivalent voltage)
• In series with a single resistor R_th (the Thevenin equivalent resistance)

        R_th
  A o--/\/\/\--+
               |
              (+)
              V_th
              (-)
               |
  B o----------+

This equivalent produces the same voltage-current relationship at terminals A-B as the original network, for any load connected there. "Linear" is the key assumption — resistors, independent sources, and linear dependent sources qualify; diodes, transistors in nonlinear regions, and other nonlinear elements do not (though the theorem still applies to a linearized small-signal model of such circuits).

Finding V_th and R_th

V_th is the open-circuit voltage at terminals A-B — remove the load, compute (or measure) the voltage across the open terminals using any standard technique (nodal analysis, mesh analysis, superposition).

R_th is the equivalent resistance seen looking into terminals A-B with all independent sources deactivated:

• Voltage sources → replaced by short circuits (0 V)
• Current sources → replaced by open circuits (0 A)
• Then combine the remaining resistors using series/parallel reduction

If the circuit contains dependent sources, you cannot simply zero them out and combine resistors, because their value depends on a circuit variable. Instead, use one of these methods:

1. Test source method: zero the independent sources, apply a 1 A test current (or 1 V test source) at terminals A-B, solve for the resulting voltage (or current), and compute R_th = V_test / I_test.
2. Short-circuit current method: compute V_th (open-circuit voltage) and I_sc (short-circuit current, load replaced by a wire), then R_th = V_th / I_sc.

The short-circuit current method is often the fastest path when the circuit already has both an open-circuit and short-circuit condition that are easy to analyze — and it doubles as the bridge to Norton's theorem, since I_sc is exactly the Norton equivalent current.

Worked Example

Consider a resistive network feeding a load at terminals A-B:

       R1=6Ω        R2=3Ω
  +----/\/\/\---+---/\/\/\---+ A
  |             |            |
 12V           R3=6Ω       (load,
  |             |           removed for
  +-------------+---------- + B

Here, a 12 V source drives R1 (6 Ω) in series with the parallel combination of R3 (6 Ω) and the branch leading to terminal A through R2 (3 Ω), with terminals A-B open (load removed).

Step 1 — Find V_th (open-circuit voltage at A-B).

With the load removed, no current flows through R2 (open branch, no path to complete a loop through B beyond the open terminals), so the voltage at node A equals the voltage at the node between R1 and R3 (call it node N), since R2 carries no current and thus drops no voltage.

Node N is a simple voltage divider between R1 and R3, both referenced to the 12 V source and ground (B):

V_N = 12 V × R3 / (R1 + R3) = 12 × 6 / (6 + 6) = 12 × 0.5 = 6 V

Since no current flows through R2, V_th = V_N = 6 V.

Step 2 — Find R_th (deactivate the 12 V source → short circuit).

Looking into terminals A-B with the source shorted: R1 and R3 are now both connected between node N and ground, in parallel with each other, and R2 connects node N to terminal A in series.

R1 ∥ R3 = (6 × 6) / (6 + 6) = 36/12 = 3 Ω
R_th = R2 + (R1 ∥ R3) = 3 + 3 = 6 Ω

Thevenin equivalent: V_th = 6 V, R_th = 6 Ω.

Verification — reattach a load and cross-check.

Suppose we reattach a 6 Ω load at A-B. Using the Thevenin equivalent:

I_load = V_th / (R_th + R_load) = 6 / (6 + 6) = 0.5 A
V_load = I_load × R_load = 0.5 × 6 = 3 V

Now check against the original circuit directly. With the 6 Ω load connected at A-B, R2 (3 Ω) is in series with the load (6 Ω) = 9 Ω, and that combination is in parallel with R3 (6 Ω):

(R2 + R_load) ∥ R3 = (9 × 6) / (9 + 6) = 54/15 = 3.6 Ω

This 3.6 Ω combination is in series with R1 (6 Ω) across the 12 V source:

Total R = 6 + 3.6 = 9.6 Ω
I_total = 12 / 9.6 = 1.25 A

Voltage at node N (across the 3.6 Ω parallel combination):

V_N = I_total × 3.6 = 1.25 × 3.6 = 4.5 V

This 4.5 V splits across the R2 + load branch (9 Ω) via current divider:

I_branch = V_N / (R2 + R_load) = 4.5 / 9 = 0.5 A
V_load = I_branch × R_load = 0.5 × 6 = 3 V

Both methods give I_load = 0.5 A and V_load = 3 V — the Thevenin equivalent checks out.

Why This Matters in Practice

• Load analysis without re-solving the whole circuit: once V_th and R_th are known, swapping in different load values is a one-line calculation instead of a full re-analysis.
• Maximum power transfer: the load resistance that extracts maximum power from a source equals R_th (R_load = R_th), a direct consequence of the Thevenin model — covered in depth in its own theorem.
• Sensor and driver output modeling: real-world signal sources (sensor outputs, op-amp outputs, battery packs) are routinely characterized by their Thevenin-equivalent open-circuit voltage and source resistance, which then predicts loading effects and voltage divider losses when connected to an ADC input or downstream stage.
• Bridging to Norton's theorem: V_th and R_th convert directly to a Norton current source I_N = V_th / R_th in parallel with the same R_th, useful when a current-source representation is more convenient for the next analysis step (source transformation is covered separately).
• Iterative design: in multi-stage networks, replacing an upstream stage with its Thevenin equivalent simplifies analyzing how it loads the next stage, without carrying the full source network through every subsequent calculation.

Practical Considerations

• The theorem holds only for linear circuits. Nonlinear elements must first be linearized (small-signal model) before applying Thevenin analysis around an operating point.
• R_th is frequency-independent only for pure resistive networks. In AC circuits with capacitors and inductors, the same theorem applies but R_th generalizes to a complex Thevenin impedance Z_th(jω), and both V_th and Z_th become functions of frequency.
• Dependent sources require the test-source or short-circuit method — never simply zero them out along with independent sources.
• Physical measurement: R_th can be estimated on the bench by measuring open-circuit voltage, then connecting a known load and measuring the loaded voltage; solving the resulting voltage-divider equation gives R_th without ever needing to short the actual source (which may be undesirable or unsafe for a real supply).

Key Takeaways

• Thevenin's theorem replaces any linear two-terminal network with a single voltage source V_th in series with a single resistance R_th.
• V_th is the open-circuit voltage at the terminals of interest; R_th is the resistance seen looking in with all independent sources deactivated.
• Circuits with dependent sources require the test-source method or the V_th / I_sc short-circuit method to find R_th.
• Always verify the equivalent by reattaching a load and comparing results against direct analysis of the original circuit.
• The Thevenin model underlies maximum power transfer analysis, source transformation to Norton form, and practical characterization of real signal sources and their output impedance.