Thevenin, Norton, and maximum power
Any linear circuit, seen from two terminals, behaves like a single source with a single resistance. Thevenin and Norton equivalents make complex circuits easy to connect and analyse.
Thevenin's theorem
Thevenin's theorem, published by Leon Charles Thevenin in 1883, states that any linear circuit, as seen from two terminals, is equivalent to a voltage source \(V_{Th}\) in series with a resistance \(R_{Th}\). Connecting any load to the terminals gives exactly the same voltage and current as connecting it to the original circuit.
Finding the Thevenin equivalent
\(V_{Th}\) is the open-circuit voltage between the terminals, with nothing connected. \(R_{Th}\) is the resistance seen looking into the terminals with all independent sources turned off (voltage sources shorted, current sources opened). Alternatively, \(R_{Th} = \frac{V_{oc}}{I_{sc}}\), the open-circuit voltage divided by the short-circuit current, a method that also works with dependent sources.
Norton's theorem
Norton's theorem (Edward Norton, 1926) gives the dual form: a current source \(I_N\) in parallel with a resistance \(R_N\). \(I_N\) is the short-circuit current between the terminals, and \(R_N = R_{Th}\). Thevenin and Norton forms convert into each other by source transformation: \(V_{Th} = I_N R_{Th}\).
Source transformation
A voltage source V in series with resistance R is equivalent, at its terminals, to a current source V/R in parallel with R. Repeatedly converting between forms, combining series voltage sources or parallel current sources, can simplify circuits step by step, an alternative to writing full nodal or mesh equations.
Why equivalents matter
Equivalents model real sources: a battery is approximately a voltage source with a small internal resistance, which is why its terminal voltage sags under heavy load. Sensors, amplifier outputs, and power supplies are described by Thevenin parameters, letting engineers predict how they will behave with any connected load without re-solving the whole circuit.
Maximum power transfer
A load receives the most power from a source when its resistance equals the source's Thevenin resistance: \(R_L = R_{Th}\). The maximum power is then \(P_{max} = \frac{V_{Th}^2}{4R_{Th}}\). At this point efficiency is only 50%, since half the power is lost in \(R_{Th}\). Matching matters in radio and audio links, but power grids instead aim for high efficiency with loads far larger than source resistance.
Measuring equivalents
In the laboratory, the Thevenin voltage can be measured with a high-resistance voltmeter, and the resistance found by measuring the voltage drop when a known load is connected, or, if safe, by measuring the short-circuit current. This is how a battery's internal resistance is estimated.
Worked example
A 12 V source in series with 4 \(\Omega\), with a 12 \(\Omega\) resistor across the terminals. Open-circuit voltage by voltage division: \(V_{Th} = 12 \times \frac{12}{16} = 9\) V. Turning off the source: \(R_{Th} = 4 \parallel 12 = \frac{48}{16} = 3\) \(\Omega\). A 3 \(\Omega\) load then receives the maximum power, \(\frac{9^2}{4 \times 3} = 6.75\) W; the Norton current is \(\frac{9}{3} = 3\) A.
Key takeaways
Thevenin's theorem (1883) replaces any linear two-terminal circuit with V_Th in series with R_Th. V_Th is the open-circuit voltage; R_Th is the resistance with independent sources off, or V_oc / I_sc. Norton's theorem (1926) uses I_N, the short-circuit current, in parallel with R_N = R_Th; source transformation converts between them. Equivalents model batteries, sensors, and amplifiers. Maximum power transfer occurs at R_L = R_Th, giving V_Th^2 / (4 R_Th) at 50% efficiency; grids instead prioritise efficiency. Equivalents can be measured with open-circuit and loaded tests.
Flashcard
Voltage source in series with resistance
Check your understanding
A source has a Thevenin voltage of 10 V and resistance of 5 ohms. What load resistance draws maximum power, and how much?