Picture the inside of a 9 V battery, or the output of a small power supply. From the outside you only ever see two terminals, and whatever you connect — a lamp, a sensor, a motor — behaves as if it were fed by a source with a fixed voltage and a fixed internal resistance. You never need a schematic of the chemistry inside the cell to predict how it drives a load. That everyday intuition is exactly what Thevenin's theorem turns into a precise, general tool.
This article explains what the Thevenin equivalent circuit is, how to find its two parameters, and why reducing a tangled network to a source and a resistor is one of the most useful moves in circuit analysis.
Why this calculation matters
Real circuits are rarely a single loop. A sensor front end, a voltage divider feeding an amplifier, a bridge network — each contains many sources and resistors. If you want to know what happens when you attach a load, solving the full network from scratch every time you change that load is slow and error-prone.
Thevenin's theorem says you only have to solve the network once. Replace everything behind the two output terminals with a single voltage source V_th in series with a single resistance R_th, and from then on the load sees an identical circuit. Swap the load, sweep its value, ask what resistance draws maximum power — all of it becomes a one-line calculation. The theorem is also the backbone of how engineers reason about source loading, signal attenuation, and impedance matching. It turns a system question into a two-number summary.
The core method
Thevenin's theorem applies to any linear two-terminal network — any network built from resistors, dependent and independent sources, where superposition holds. It states that, seen from the chosen pair of terminals, the network is indistinguishable from this:
V_th in series with R_th, feeding the load
Two quantities define it.
The Thevenin voltage V_th is the open-circuit voltage at the terminals. Remove the load, leave the terminals open, and measure (or compute) the voltage across them. With no load current, that voltage is V_th.
The Thevenin resistance R_th is the resistance looking back into the terminals with all independent sources deactivated. Deactivating a source has a specific meaning: short every independent voltage source (replace it with a wire) and open every independent current source (replace it with a gap). Then compute the equivalent resistance seen from the terminals.
V_th = open-circuit terminal voltage
R_th = terminal resistance with independent sources deactivated
There is a second route to R_th that works even when dependent sources are present. Find the short-circuit current I_sc that flows when you bridge the terminals with a wire, then use:
R_th = V_th / I_sc
This Ohm's-law relationship between the open-circuit voltage and the short-circuit current is itself a compact statement of the theorem. Once V_th and R_th are known, the current into any load resistance R_L is simply V_th / (R_th + R_L).
A worked example
Take a concrete network. A 12 V source connects through a 4 ohm series resistor R1 to the output node. From that node, a 6 ohm resistor R2 runs to ground. The two output terminals are the R2 node and ground. We want the Thevenin equivalent seen at those terminals.
Step 1 — find V_th, the open-circuit voltage.
With the terminals open, no current is drawn from the output, so R1 and R2 form a simple voltage divider. The voltage across R2 is the open-circuit terminal voltage:
V_th = 12 * R2 / (R1 + R2)
V_th = 12 * 6 / (4 + 6)
V_th = 12 * 6 / 10 = 7.2 V
Step 2 — find R_th, the resistance looking back in.
Deactivate the source: replace the 12 V source with a short. Now look into the terminals. R1 has one end at the output node and its other end is now grounded through the short. R2 also runs from the output node to ground. So R1 and R2 appear in parallel:
R_th = R1 * R2 / (R1 + R2)
R_th = 4 * 6 / (4 + 6)
R_th = 24 / 10 = 2.4 ohm
Result. The entire network behaves, at those two terminals, as a 7.2 V source in series with 2.4 ohm. Connect any load and you can read off the result immediately. A 2.4 ohm load, for instance, draws 7.2 / (2.4 + 2.4) = 1.5 A — and because the load matches R_th, that is the maximum-power-transfer point.
Common mistakes
Forgetting to deactivate the sources for R_th. R_th is a property of the network's resistances alone. If you leave the 12 V source active while computing the resistance, you get nonsense. Short voltage sources, open current sources, then look in.
Confusing "short" and "open" when deactivating. A voltage source becomes a short (a wire); a current source becomes an open (a gap). Reversing the two is one of the most frequent slips and quietly corrupts R_th.
Measuring V_th with the load still attached. V_th is the open-circuit voltage. If a load is connected, current flows, and the terminal voltage drops below V_th. The load must be removed — physically or on paper — before you read or compute V_th.
Applying it to a nonlinear network. Thevenin's theorem rests on linearity and superposition. A network containing a diode in its forward-conducting region, or any other nonlinear element, has no single fixed Thevenin equivalent; the equivalent would change with operating point.
Mishandling dependent sources. A dependent source is not deactivated when finding R_th. If the network has one, use the V_th / I_sc method, or apply a test source at the terminals and compute the resulting current.
Try the interactive NovaSolver calculator
Working V_th and R_th by hand is quick for a two-resistor divider, but the payoff is seeing how a load responds once the equivalent is fixed. The Thevenin Equivalent Simulator — Maximum Power Transfer on NovaSolver lets you set V_th, R_th, a wiring resistance, and a load resistance, then shows load current, load voltage, load power, and efficiency updating in real time — so you can watch the power curve peak exactly where the load equals the source resistance.
Related calculators
- RC/RL Circuit Transient Response Analyzer — once a network is reduced to a source and a resistance, this shows how it charges a capacitor or energizes an inductor over time.
- Coulomb Law Simulator — to step back to the electrostatic force that underlies every voltage and current in the first place.
- Series RLC Circuit Simulator — extends the same equivalent-circuit thinking from DC resistance to AC impedance and resonance.
You can browse the full set in the electromagnetics tools hub.
Closing note
Thevenin's theorem is a small idea with enormous reach. Any linear two-terminal network, no matter how many sources and resistors it hides, collapses into one voltage and one resistance the moment you fix the terminals. Find V_th from the open circuit, find R_th with the sources deactivated, and every load question after that is a single division. Master the two-step procedure, keep the deactivation rules straight, and the most intimidating networks become as simple as a battery with an internal resistance.
Top comments (0)