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Thévenin's theorem

Thévenin's theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Thévenin's theorem rather than just read about it. In short: As originally stated in terms of direct-current resistive circuits only, Thévenin's theorem states that "Any linear electrical network containing only voltage sources, current sources and resistances can be replaced at terminals A–B by an equivalent combination of a voltage source Vth in a series connection with a resistance Rth." The equivalent voltage Vth is the voltage obtained at terminals A–B of the network wit…

Thévenin's theorem — main illustration
Thévenin's theorem — illustration

Key takeaways

  • Thévenin's theorem belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Thévenin's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Thévenin's theorem from memory before moving on to harder problems.

Reference excerpt

As originally stated in terms of direct-current resistive circuits only, Thévenin's theorem states that "Any linear electrical network containing only voltage sources, current sources and resistances can be replaced at terminals A–B by an equivalent combination of a voltage source Vth in a series connection with a resistance Rth."

The equivalent voltage Vth is the voltage obtained at terminals A–B of the network with terminals A–B open circuited. The equivalent resistance Rth is the resistance that the circuit between terminals A and B would have if all ideal voltage sources in the circuit were replaced by a short circuit and all ideal current sources were replaced by an open circuit (i.e., the sources are set to provide zero voltages and currents). If terminals A and B are connected to one another (short), then the current flowing from A and B will be V t h R t h {\textstyle {\frac {V_{\mathrm {th} }}{R_{\mathrm {th} }}}} according to the Thévenin equivalent circuit. This means that Rth could alternatively be calculated as Vth divided by the short-circuit current between A and B when they are connected together. In circuit theory terms, the theorem allows any one-port network to be reduced to a single voltage source and a single impedance. The theorem also applies to frequency domain AC circuits consisting of reactive (inductive and capacitive) and resistive impedances. It means the theorem applies for AC in an exactly same way to DC except that resistances are generalized to impedances. The theorem was independently derived in 1853 by the German scientist Hermann von Helmholtz and in 1883 by Léon Charles Thévenin (1857–1926), an electrical engineer with France's national Postes et Télégraphes telecommunications organization. Thévenin's theorem and its dual, Norton's theorem, are widely used to make circuit analysis simpler and to study a circuit's initial-condition and steady-state response. Thévenin's theorem can be used to convert any circuit's sources and impedances to a Thévenin equivalent; use of the theorem may in some cases be more convenient than use of Kirchhoff's circuit laws.

A proof of the theorem Various proofs have been given of Thévenin's theorem. Perhaps the simplest of these was the proof in Thévenin's original paper. A consensus exists that Thévenin's proof is both correct and general in its applicability. The proof goes as follows: Consider an active network containing impedances, (constant-) voltage sources and (constant-) current sources. The configuration of the network can be anything. Access to the network is provided by a pair of terminals. Designate the voltage measured between the terminals as Vθ, as shown in the box on the left side of Figure 2.

Suppose that the voltage sources within the box are replaced by short circuits, and the current sources by open circuits. If this is done, no voltage appears across the terminals, and it is possible to measure the impedance between the terminals. Call this impedance Zθ. Now suppose that one attaches some linear network to the terminals of the box, having impedance Ze, as in Figure 2a. We wish to find the current I through Ze. The answer is not obvious, since the terminal voltage will not be Vθ after Ze is connected. Instead, we imagine that we attach, in series with impedance Ze, a source with electromotive force E equal to Vθ but directed to oppose Vθ, as shown in Figure 2b. No current will then flow through Ze since E balances Vθ. Next, we insert another source of electromotive force, E1, in series with Ze, where E1 has the same magnitude as E but is opposed in direction (see Figure 2c). The current, I1, can be determined as follows: it is the current that would result from E1 acting alone, with all other sources (within the active network and the external network) set to zero. This current is, therefore,

I 1 = E 1 Z e + Z θ = V θ Z e + Z θ {\displaystyle I_{1}={\frac {E_{1}}{Z_{e}+Z_{\theta }}}={\frac {V_{\theta }}{Z_{e}+Z_{\theta }}}} because Ze is the impedance external to the box and Zθ looking into the box when its sources are zero. Finally, we note that E and E1 can be removed together without changing the current, and when they are removed, we are back to Figure 2a. Therefore, I1 is the current, I, that we are seeking, i.e.

I = V θ Z e + Z θ {\displaystyle I={\frac {V_{\theta }}{Z_{e}+Z_{\theta }}}}

thus, completing the proof. Figure 2d shows the Thévenin equivalent circuit.

… excerpt ends here. Continue reading the full article.

Illustrations

Thévenin's theorem: Fig. 1. Any black box containing only resistances, voltage sources and current sources, can be replaced by a Thévenin equivalent circuit consisting of an equivalent voltage source in series connection with an equivalent resistance.
Fig. 1. Any black box containing only resistances, voltage sources and current sources, can be replaced by a Thévenin equivalent circuit consisting of an equivalent voltage source in series connection with an equivalent resistance.
Thévenin's theorem: Fig. 2. Figure used in the proof of Thévenin's theorem.
Fig. 2. Figure used in the proof of Thévenin's theorem.
Thévenin's theorem: Fig. 3. Original circuitThe equivalent voltageThe equivalent resistanceThe equivalent circuit
Fig. 3. Original circuitThe equivalent voltageThe equivalent resistanceThe equivalent circuit
Thévenin's theorem: Fig. 4. Norton-Thevenin conversion
Fig. 4. Norton-Thevenin conversion

Worked examples

Example 1 — a first encounter with Thévenin's theorem

Start with the simplest possible case. Write down what Thévenin's theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Thévenin's theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Thévenin's theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Thévenin's theorem

In research
Thévenin's theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Thévenin's theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Thévenin's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Circuit theorems, Linear electronic circuits, so understanding it makes those chapters shorter.
In everyday life
Look for Thévenin's theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Thévenin's theorem in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Thévenin's theorem means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Thévenin's theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Thévenin's theorem in simple terms?

As originally stated in terms of direct-current resistive circuits only, Thévenin's theorem states that "Any linear electrical network containing only voltage sources, current sources and resistances can be replaced at terminals A–B by an equivalent combination of a voltage source Vth in a series c…

Why does Thévenin's theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Thévenin's theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Thévenin's theorem.

Tags

  • Circuit theorems
  • Linear electronic circuits

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